id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
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proofwiki-18900 | Integral of Horizontal Section of Measurable Function gives Measurable Function | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be $\sigma$-finite measure spaces.
Let $f : X \times Y \to \overline \R_{\ge 0}$ be a positive $\Sigma_X \otimes \Sigma_Y$-measurable function, where $\Sigma_X \otimes \Sigma_Y$ is the product $\sigma$-algebra of $\Sigma_X$ and $\Sigma_Y$.
Define the fun... | First we prove the case of:
:$f = \chi_E$
where $E$ is a $\Sigma_X \otimes \Sigma_Y$-measurable set.
From Horizontal Section of Characteristic Function is Characteristic Function of Horizontal Section, we have:
:$f^y = \chi_{E^y}$
From Horizontal Section of Measurable Function is Measurable, we also have:
:$f^y$ is $... | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be [[Definition:Sigma-Finite Measure Space|$\sigma$-finite]] [[Definition:Measure Space|measure spaces]].
Let $f : X \times Y \to \overline \R_{\ge 0}$ be a [[Definition:Positive Measurable Function|positive $\Sigma_X \otimes \Sigma_Y$-measurable functi... | First we prove the case of:
:$f = \chi_E$
where $E$ is a [[Definition:Measurable Set|$\Sigma_X \otimes \Sigma_Y$-measurable set]].
From [[Horizontal Section of Characteristic Function is Characteristic Function of Horizontal Section]], we have:
:$f^y = \chi_{E^y}$
From [[Horizontal Section of Measurable Function ... | Integral of Horizontal Section of Measurable Function gives Measurable Function | https://proofwiki.org/wiki/Integral_of_Horizontal_Section_of_Measurable_Function_gives_Measurable_Function | https://proofwiki.org/wiki/Integral_of_Horizontal_Section_of_Measurable_Function_gives_Measurable_Function | [
"Measurable Functions",
"Horizontal Section of Functions"
] | [
"Definition:Sigma-Finite Measure Space",
"Definition:Measure Space",
"Definition:Measurable Function/Positive",
"Definition:Product Sigma-Algebra",
"Definition:Extended Real-Valued Function",
"Definition:Horizontal Section of Function",
"Definition:Measurable Function"
] | [
"Definition:Measurable Set",
"Horizontal Section of Characteristic Function is Characteristic Function of Horizontal Section",
"Horizontal Section of Measurable Function is Measurable",
"Definition:Measurable Function",
"Definition:Integral of Positive Measurable Function",
"Integral of Characteristic Fun... |
proofwiki-18901 | Horizontal Section of Measurable Function is Measurable | Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be measurable spaces.
Let $f : X \times Y \to \overline \R$ be a $\Sigma_X \otimes \Sigma_Y$-measurable function where $\Sigma_X \otimes \Sigma_Y$ is the product $\sigma$-algebra of $\Sigma_X$ and $\Sigma_Y$.
Let $y \in Y$.
Then:
:$f^y$ is $\Sigma_X$-measurable
... | By the definition of a $\Sigma_X$-measurable function, we have that:
:$f^{-1} \sqbrk D \in \Sigma_X \otimes \Sigma_Y$ for each Borel set $D \subseteq \R$.
We aim to show that:
:$\paren {f^y}^{-1} \sqbrk D \in \Sigma_X$ for each Borel set $D \subseteq \R$.
Let $D \subseteq \R$ be a Borel set.
From Preimage of Horizon... | Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be [[Definition:Measurable Space|measurable spaces]].
Let $f : X \times Y \to \overline \R$ be a [[Definition:Measurable Function|$\Sigma_X \otimes \Sigma_Y$-measurable function]] where $\Sigma_X \otimes \Sigma_Y$ is the [[Definition:Product Sigma-Algebra|product... | By the definition of a [[Definition:Measurable Function|$\Sigma_X$-measurable function]], we have that:
:$f^{-1} \sqbrk D \in \Sigma_X \otimes \Sigma_Y$ for each [[Definition:Borel Set|Borel set]] $D \subseteq \R$.
We aim to show that:
:$\paren {f^y}^{-1} \sqbrk D \in \Sigma_X$ for each [[Definition:Borel Set|Bore... | Horizontal Section of Measurable Function is Measurable | https://proofwiki.org/wiki/Horizontal_Section_of_Measurable_Function_is_Measurable | https://proofwiki.org/wiki/Horizontal_Section_of_Measurable_Function_is_Measurable | [
"Horizontal Section of Functions",
"Measurable Functions"
] | [
"Definition:Measurable Space",
"Definition:Measurable Function",
"Definition:Product Sigma-Algebra",
"Definition:Measurable Function",
"Definition:Horizontal Section of Function"
] | [
"Definition:Measurable Function",
"Definition:Borel Sigma-Algebra/Borel Set",
"Definition:Borel Sigma-Algebra/Borel Set",
"Definition:Borel Sigma-Algebra/Borel Set",
"Preimage of Horizontal Section of Function is Horizontal Section of Preimage",
"Horizontal Section of Measurable Set is Measurable",
"Def... |
proofwiki-18902 | Vertical Section of Measurable Function is Measurable | Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be measurable spaces.
Let $f : X \times Y \to \overline \R$ be a $\Sigma_X \otimes \Sigma_Y$-measurable function where $\Sigma_X \otimes \Sigma_Y$ is the product $\sigma$-algebra of $\Sigma_X$ and $\Sigma_Y$.
Let $x \in X$.
Then:
:$f_x$ is $\Sigma_Y$-measurable
... | By the definition of a $\Sigma_X$-measurable function, we have that:
:$f^{-1} \sqbrk D \in \Sigma_X \otimes \Sigma_Y$ for each Borel set $D \subseteq \R$.
We aim to show that:
:$\paren {f_x}^{-1} \sqbrk D \in \Sigma_Y$ for each Borel set $D \subseteq \R$.
Let $D \subseteq \R$ be a Borel set.
From Preimage of Vertica... | Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be [[Definition:Measurable Space|measurable spaces]].
Let $f : X \times Y \to \overline \R$ be a [[Definition:Measurable Function|$\Sigma_X \otimes \Sigma_Y$-measurable function]] where $\Sigma_X \otimes \Sigma_Y$ is the [[Definition:Product Sigma-Algebra|product... | By the definition of a [[Definition:Measurable Function|$\Sigma_X$-measurable function]], we have that:
:$f^{-1} \sqbrk D \in \Sigma_X \otimes \Sigma_Y$ for each [[Definition:Borel Set|Borel set]] $D \subseteq \R$.
We aim to show that:
:$\paren {f_x}^{-1} \sqbrk D \in \Sigma_Y$ for each [[Definition:Borel Set|Bore... | Vertical Section of Measurable Function is Measurable | https://proofwiki.org/wiki/Vertical_Section_of_Measurable_Function_is_Measurable | https://proofwiki.org/wiki/Vertical_Section_of_Measurable_Function_is_Measurable | [
"Vertical Section of Functions",
"Measurable Functions"
] | [
"Definition:Measurable Space",
"Definition:Measurable Function",
"Definition:Product Sigma-Algebra",
"Definition:Measurable Function",
"Definition:Vertical Section of Function"
] | [
"Definition:Measurable Function",
"Definition:Borel Sigma-Algebra/Borel Set",
"Definition:Borel Sigma-Algebra/Borel Set",
"Definition:Borel Sigma-Algebra/Borel Set",
"Preimage of Vertical Section of Function is Vertical Section of Preimage",
"Horizontal Section of Measurable Set is Measurable",
"Definit... |
proofwiki-18903 | Preimage of Vertical Section of Function is Vertical Section of Preimage | Let $X$ and $Y$ be sets.
Let $f : X \times Y \to \overline \R$ be an extended real-valued function.
Let $x \in X$.
Let $D \subseteq \R$.
Then:
:$\paren {f_x}^{-1} \sqbrk D = \paren {f^{-1} \sqbrk D}_x$
where:
:$f_x$ is the $x$-vertical section of $f$
:$\paren {f^{-1} \sqbrk D}_x$ is the $x$-vertical section of $f^{-... | Note that:
:$y \in \paren {f_x}^{-1} \sqbrk D$
{{iff}}:
:$\map {f_x} y \in D$
from the definition of preimage.
That is, by the definition of the $x$-vertical section:
:$\map f {x, y} \in D$
From the definition of preimage, this is equivalent to:
:$\paren {x, y} \in f^{-1} \sqbrk D$
Which in turn is equivalent to:
:... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $f : X \times Y \to \overline \R$ be an [[Definition:Extended Real-Valued Function|extended real-valued function]].
Let $x \in X$.
Let $D \subseteq \R$.
Then:
:$\paren {f_x}^{-1} \sqbrk D = \paren {f^{-1} \sqbrk D}_x$
where:
:$f_x$ is the [[Definition:Vertical S... | Note that:
:$y \in \paren {f_x}^{-1} \sqbrk D$
{{iff}}:
:$\map {f_x} y \in D$
from the definition of [[Definition:Preimage of Mapping|preimage]].
That is, by the definition of the [[Definition:Vertical Section of Function|$x$-vertical section]]:
:$\map f {x, y} \in D$
From the definition of [[Definition:Preima... | Preimage of Vertical Section of Function is Vertical Section of Preimage | https://proofwiki.org/wiki/Preimage_of_Vertical_Section_of_Function_is_Vertical_Section_of_Preimage | https://proofwiki.org/wiki/Preimage_of_Vertical_Section_of_Function_is_Vertical_Section_of_Preimage | [
"Vertical Section of Functions",
"Vertical Section of Sets",
"Preimages under Mappings"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Vertical Section of Function",
"Definition:Vertical Section of Set"
] | [
"Definition:Preimage/Mapping/Mapping",
"Definition:Vertical Section of Function",
"Definition:Preimage/Mapping/Mapping",
"Definition:Vertical Section of Function",
"Category:Vertical Section of Functions",
"Category:Vertical Section of Sets",
"Category:Preimages under Mappings"
] |
proofwiki-18904 | Measure of Vertical Section of Cartesian Product | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be measure spaces.
Let $S_1 \in \Sigma_X$ and $S_2 \in \Sigma_Y$.
Let $x \in X$.
Then:
:$\map {\nu} {\paren {S_1 \times S_2}_x} = \map {\nu} {S_2} \map {\chi_{S_1} } x$
where:
:$\paren {S_1 \times S_2}_x$ is the $x$-vertical section of $S_1 \times S_2... | From Vertical Section of Cartesian Product, we have:
:$\paren {S_1 \times S_2}_x = \begin{cases}S_2 & x \in S_1 \\ \O & x \not \in S_1\end{cases}$
So:
:$\map {\nu} {\paren {S_1 \times S_2}_x} = \begin{cases}\map {\nu} {S_2} & x \in S_1 \\ 0 & x \not \in S_1\end{cases}$
That is:
:$\map {\nu} {\paren {S_1 \times S_2}_x... | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be [[Definition:Measure Space|measure spaces]].
Let $S_1 \in \Sigma_X$ and $S_2 \in \Sigma_Y$.
Let $x \in X$.
Then:
:$\map {\nu} {\paren {S_1 \times S_2}_x} = \map {\nu} {S_2} \map {\chi_{S_1} } x$
where:
:$\paren {S_1 \times S_2}_x$ is the [[D... | From [[Vertical Section of Cartesian Product]], we have:
:$\paren {S_1 \times S_2}_x = \begin{cases}S_2 & x \in S_1 \\ \O & x \not \in S_1\end{cases}$
So:
:$\map {\nu} {\paren {S_1 \times S_2}_x} = \begin{cases}\map {\nu} {S_2} & x \in S_1 \\ 0 & x \not \in S_1\end{cases}$
That is:
:$\map {\nu} {\paren {S_1 \tim... | Measure of Vertical Section of Cartesian Product | https://proofwiki.org/wiki/Measure_of_Vertical_Section_of_Cartesian_Product | https://proofwiki.org/wiki/Measure_of_Vertical_Section_of_Cartesian_Product | [
"Vertical Section of Sets",
"Cartesian Product"
] | [
"Definition:Measure Space",
"Definition:Vertical Section of Set",
"Definition:Characteristic Function (Set Theory)"
] | [
"Vertical Section of Cartesian Product",
"Definition:Characteristic Function (Set Theory)",
"Category:Vertical Section of Sets",
"Category:Cartesian Product"
] |
proofwiki-18905 | Measure of Horizontal Section of Cartesian Product | Let $\struct {X, \Sigma_X, \mu_X}$ and $\struct {Y, \Sigma_Y, \mu_Y}$ be measure spaces.
Let $S_1 \in \Sigma_X$ and $S_2 \in \Sigma_Y$.
Let $y \in Y$.
Then:
:$\map {\mu_X} {\paren {S_1 \times S_2}^y} = \map {\mu_X} {S_1} \map {\chi_{S_2} } y$
where:
:$\paren {S_1 \times S_2}^y$ is the $y$-horizontal section of $S_1 ... | From Horizontal Section of Cartesian Product, we have:
:$\paren {S_1 \times S_2}^y = \begin{cases}S_1 & y \in S_2 \\ \O & y \not \in S_2\end{cases}$
So:
:$\map {\mu_X} {\paren {S_1 \times S_2}^y} = \begin{cases}\map {\mu_X} {S_1} & y \in S_2 \\ 0 & y \not \in S_2\end{cases}$
That is:
:$\map {\mu_X} {\paren {S_1 \time... | Let $\struct {X, \Sigma_X, \mu_X}$ and $\struct {Y, \Sigma_Y, \mu_Y}$ be [[Definition:Measure Space|measure spaces]].
Let $S_1 \in \Sigma_X$ and $S_2 \in \Sigma_Y$.
Let $y \in Y$.
Then:
:$\map {\mu_X} {\paren {S_1 \times S_2}^y} = \map {\mu_X} {S_1} \map {\chi_{S_2} } y$
where:
:$\paren {S_1 \times S_2}^y$ is... | From [[Horizontal Section of Cartesian Product]], we have:
:$\paren {S_1 \times S_2}^y = \begin{cases}S_1 & y \in S_2 \\ \O & y \not \in S_2\end{cases}$
So:
:$\map {\mu_X} {\paren {S_1 \times S_2}^y} = \begin{cases}\map {\mu_X} {S_1} & y \in S_2 \\ 0 & y \not \in S_2\end{cases}$
That is:
:$\map {\mu_X} {\paren {... | Measure of Horizontal Section of Cartesian Product | https://proofwiki.org/wiki/Measure_of_Horizontal_Section_of_Cartesian_Product | https://proofwiki.org/wiki/Measure_of_Horizontal_Section_of_Cartesian_Product | [
"Horizontal Section of Sets",
"Cartesian Product"
] | [
"Definition:Measure Space",
"Definition:Horizontal Section of Set",
"Definition:Characteristic Function (Set Theory)"
] | [
"Horizontal Section of Cartesian Product",
"Definition:Characteristic Function (Set Theory)",
"Category:Horizontal Section of Sets",
"Category:Cartesian Product"
] |
proofwiki-18906 | Intersection of Horizontal Sections is Horizontal Section of Intersection | Let $X$ and $Y$ be sets.
Let $\set {E_\alpha : \alpha \mathop \in A}$ be a set of subsets of $X \times Y$.
Let $y \in Y$.
Then:
:$\ds \paren {\bigcap_{\alpha \mathop \in A} E_\alpha}^y = \bigcap_{\alpha \mathop \in A} \paren {E_\alpha}^y$
where:
:$\ds \paren {\bigcap_{\alpha \mathop \in A} E_\alpha}^y$ is the $y$-ho... | Note that:
:$\ds x \in \bigcap_{\alpha \mathop \in A} \paren {E_\alpha}^y$
{{iff}}:
:$x \in \paren {E_\alpha}^y$ for all $\alpha \in A$.
From the definition of the $x$-horizontal section, this is equivalent to:
:$\tuple {x, y} \in E_\alpha$ for all $\alpha \in A$.
This in turn is equivalent to:
:$\ds \tuple {x, y} \... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $\set {E_\alpha : \alpha \mathop \in A}$ be a [[Definition:Set|set]] of subsets of $X \times Y$.
Let $y \in Y$.
Then:
:$\ds \paren {\bigcap_{\alpha \mathop \in A} E_\alpha}^y = \bigcap_{\alpha \mathop \in A} \paren {E_\alpha}^y$
where:
:$\ds \paren {\bigcap_{\alp... | Note that:
:$\ds x \in \bigcap_{\alpha \mathop \in A} \paren {E_\alpha}^y$
{{iff}}:
:$x \in \paren {E_\alpha}^y$ for all $\alpha \in A$.
From the definition of the [[Definition:Horizontal Section of Set|$x$-horizontal section]], this is equivalent to:
:$\tuple {x, y} \in E_\alpha$ for all $\alpha \in A$.
This i... | Intersection of Horizontal Sections is Horizontal Section of Intersection | https://proofwiki.org/wiki/Intersection_of_Horizontal_Sections_is_Horizontal_Section_of_Intersection | https://proofwiki.org/wiki/Intersection_of_Horizontal_Sections_is_Horizontal_Section_of_Intersection | [
"Set Intersection",
"Horizontal Section of Sets"
] | [
"Definition:Set",
"Definition:Set",
"Definition:Horizontal Section of Set",
"Definition:Horizontal Section of Set"
] | [
"Definition:Horizontal Section of Set",
"Definition:Horizontal Section of Set",
"Category:Set Intersection",
"Category:Horizontal Section of Sets"
] |
proofwiki-18907 | Vertical Section of Empty Set | Let $X$ and $Y$ be sets.
Let $x \in X$.
Then:
:$\O_x = \O$
where $\O$ is the empty set and $\O_x$ is the $x$-vertical section of $\O$. | {{AimForCont}} suppose that:
:$y \in \O_x$
Then from the definition of the $x$-vertical section, we have:
:$\tuple {x, y} \in \O$
This is impossible from the definition of the empty set.
So:
:there exists no $y \in \O_x$
giving:
:$\O_x = \O$
{{qed}}
Category:Vertical Section of Sets
lmoohqlk21jxeb460jwac7p71ynwtis | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $x \in X$.
Then:
:$\O_x = \O$
where $\O$ is the [[Definition:Empty Set|empty set]] and $\O_x$ is the [[Definition:Vertical Section of Set|$x$-vertical section]] of $\O$. | {{AimForCont}} suppose that:
:$y \in \O_x$
Then from the definition of the [[Definition:Vertical Section of Set|$x$-vertical section]], we have:
:$\tuple {x, y} \in \O$
This is impossible from the definition of the [[Definition:Empty Set|empty set]].
So:
:there exists no $y \in \O_x$
giving:
:$\O_x = \O$
{{qe... | Vertical Section of Empty Set | https://proofwiki.org/wiki/Vertical_Section_of_Empty_Set | https://proofwiki.org/wiki/Vertical_Section_of_Empty_Set | [
"Vertical Section of Sets"
] | [
"Definition:Set",
"Definition:Empty Set",
"Definition:Vertical Section of Set"
] | [
"Definition:Vertical Section of Set",
"Definition:Empty Set",
"Category:Vertical Section of Sets"
] |
proofwiki-18908 | Horizontal Section of Empty Set | Let $X$ and $Y$ be sets.
Let $y \in Y$.
Then:
:$\O^y = \O$
where $\O$ is the empty set and $\O^y$ is the $y$-horizontal section of $\O$. | {{AimForCont}} suppose that:
:$x \in \O^y$
Then from the definition of the $x$-vertical section, we have:
:$\tuple {x, y} \in \O$
This is impossible from the definition of the empty set.
So:
:there exists no $x \in \O^y$
giving:
:$\O_x = \O$
{{qed}}
Category:Horizontal Section of Sets
lwr8z5v192p5xwbd7io581o0wedbi81 | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $y \in Y$.
Then:
:$\O^y = \O$
where $\O$ is the [[Definition:Empty Set|empty set]] and $\O^y$ is the [[Definition:Horizontal Section of Set|$y$-horizontal section]] of $\O$. | {{AimForCont}} suppose that:
:$x \in \O^y$
Then from the definition of the [[Definition:Vertical Section of Set|$x$-vertical section]], we have:
:$\tuple {x, y} \in \O$
This is impossible from the definition of the [[Definition:Empty Set|empty set]].
So:
:there exists no $x \in \O^y$
giving:
:$\O_x = \O$
{{qe... | Horizontal Section of Empty Set | https://proofwiki.org/wiki/Horizontal_Section_of_Empty_Set | https://proofwiki.org/wiki/Horizontal_Section_of_Empty_Set | [
"Horizontal Section of Sets"
] | [
"Definition:Set",
"Definition:Empty Set",
"Definition:Horizontal Section of Set"
] | [
"Definition:Vertical Section of Set",
"Definition:Empty Set",
"Category:Horizontal Section of Sets"
] |
proofwiki-18909 | Measure of Horizontal Section of Measurable Set gives Measurable Function | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be $\sigma$-finite measure spaces.
For each $E \in \Sigma_X \otimes \Sigma_Y$, define the function $f_E : Y \to \overline \R$ by:
:$\map {f_E} x = \map {\mu} {E^y}$
for each $y \in Y$ where:
:$\Sigma_X \otimes \Sigma_Y$ is the product $\sigma$-algebra ... | From Horizontal Section of Measurable Set is Measurable, the function $f_E$ is certainly well-defined for each $E \in \Sigma_X \otimes \Sigma_Y$.
First suppose that $\mu$ is a finite measure.
Let:
:$\mathcal F = \set {E \in \Sigma_X \otimes \Sigma_Y : f_E \text { is } \Sigma_Y\text{-measurable} }$
We aim to show that... | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be [[Definition:Sigma-Finite Measure Space|$\sigma$-finite]] [[Definition:Measure Space|measure spaces]].
For each $E \in \Sigma_X \otimes \Sigma_Y$, define the [[Definition:Extended Real-Valued Function|function]] $f_E : Y \to \overline \R$ by:
:$\ma... | From [[Horizontal Section of Measurable Set is Measurable]], the [[Definition:Extended Real-Valued Function|function]] $f_E$ is certainly well-defined for each $E \in \Sigma_X \otimes \Sigma_Y$.
First suppose that $\mu$ is a [[Definition:Finite Measure|finite measure]].
Let:
:$\mathcal F = \set {E \in \Sigma_X \o... | Measure of Horizontal Section of Measurable Set gives Measurable Function | https://proofwiki.org/wiki/Measure_of_Horizontal_Section_of_Measurable_Set_gives_Measurable_Function | https://proofwiki.org/wiki/Measure_of_Horizontal_Section_of_Measurable_Set_gives_Measurable_Function | [
"Horizontal Section of Sets"
] | [
"Definition:Sigma-Finite Measure Space",
"Definition:Measure Space",
"Definition:Extended Real-Valued Function",
"Definition:Product Sigma-Algebra",
"Definition:Horizontal Section of Set",
"Definition:Measurable Function"
] | [
"Horizontal Section of Measurable Set is Measurable",
"Definition:Extended Real-Valued Function",
"Definition:Finite Measure",
"Definition:Measurable Function",
"Measure of Horizontal Section of Cartesian Product",
"Definition:Measurable Set",
"Definition:Measurable Function",
"Pointwise Scalar Multip... |
proofwiki-18910 | Pointwise Sum of Measurable Functions is Measurable/General Result | Let $\sequence {\alpha_n}_{n \mathop \in \N}$ be a sequence of real numbers.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence of $\Sigma$-measurable functions $f_n : X \to \overline \R$ such that:
:for each $N \in \N$ and $x \in X$, the sum $\ds \sum_{n \mathop = 1}^N \alpha_n \map {f_n} x$ is well-defined.
Then... | We proceed by induction.
For all $N \in \N$ let $\map P N$ be the proposition:
:$\ds \sum_{n \mathop = 1}^N \alpha_n f_n$ is $\Sigma$-measurable. | Let $\sequence {\alpha_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Number|real numbers]].
Let $\sequence {f_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Measurable Function|$\Sigma$-measurable]] [[Definition:Extended Real Valued Function|functions]]... | We proceed by [[Principle of Mathematical Induction|induction]].
For all $N \in \N$ let $\map P N$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{n \mathop = 1}^N \alpha_n f_n$ is [[Definition:Measurable Function|$\Sigma$-measurable]]. | Pointwise Sum of Measurable Functions is Measurable/General Result | https://proofwiki.org/wiki/Pointwise_Sum_of_Measurable_Functions_is_Measurable/General_Result | https://proofwiki.org/wiki/Pointwise_Sum_of_Measurable_Functions_is_Measurable/General_Result | [
"Pointwise Sum of Measurable Functions is Measurable"
] | [
"Definition:Sequence",
"Definition:Real Number",
"Definition:Sequence",
"Definition:Measurable Function",
"Definition:Extended Real-Valued Function",
"Definition:Summation",
"Definition:Measurable Function"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Measurable Function... |
proofwiki-18911 | Vertical Section of Linear Combination of Functions is Linear Combination of Vertical Sections | Let $X$ and $Y$ be sets.
Let $f_1, f_2, \ldots, f_n : X \times Y \to \overline \R$ be functions.
Let $\alpha_1, \alpha_2, \ldots, \alpha_n$ be real numbers.
Let $x \in X$.
Then:
:$\ds \paren {\sum_{k \mathop = 1}^n \alpha_k f_k}_x = \sum_{k \mathop = 1}^n \alpha_k \paren {f_k}_x$
where $f_x$ denotes the $x$-vertical... | Let $y \in Y$.
We have:
{{begin-eqn}}
{{eqn | l = \map {\paren {\sum_{k \mathop = 1}^n \alpha_k f_k}_x} y
| r = \map {\paren {\sum_{k \mathop = 1}^n \alpha_k f_k} } {x, y}
| c = {{Defof|Vertical Section of Function}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \alpha_k \map {f_k} {x, y}
}}
{{eqn | r = \sum_{k \mathop = ... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $f_1, f_2, \ldots, f_n : X \times Y \to \overline \R$ be [[Definition:Extended Real-Valued Function|functions]].
Let $\alpha_1, \alpha_2, \ldots, \alpha_n$ be [[Definition:Real Number|real numbers]].
Let $x \in X$.
Then:
:$\ds \paren {\sum_{k \mathop = 1}^n \alph... | Let $y \in Y$.
We have:
{{begin-eqn}}
{{eqn | l = \map {\paren {\sum_{k \mathop = 1}^n \alpha_k f_k}_x} y
| r = \map {\paren {\sum_{k \mathop = 1}^n \alpha_k f_k} } {x, y}
| c = {{Defof|Vertical Section of Function}}
}}
{{eqn | r = \sum_{k \mathop = 1}^n \alpha_k \map {f_k} {x, y}
}}
{{eqn | r = \sum_{k \mathop ... | Vertical Section of Linear Combination of Functions is Linear Combination of Vertical Sections | https://proofwiki.org/wiki/Vertical_Section_of_Linear_Combination_of_Functions_is_Linear_Combination_of_Vertical_Sections | https://proofwiki.org/wiki/Vertical_Section_of_Linear_Combination_of_Functions_is_Linear_Combination_of_Vertical_Sections | [
"Vertical Section of Functions"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Real Number",
"Definition:Vertical Section of Function",
"Definition:Extended Real-Valued Function"
] | [
"Category:Vertical Section of Functions"
] |
proofwiki-18912 | Horizontal Section of Simple Function is Simple Function | Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be measurable spaces.
Let $\struct {X \times Y, \Sigma_X \otimes \Sigma_Y}$ be the product measurable space of $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$.
Let $f : X \times Y \to \R$ be a simple function.
Let $y \in Y$.
Then $f^y : X \to \R$ is a simple ... | Write the standard representation of $f$ as:
:$\ds f = \sum_{k \mathop = 1}^n a_k \chi_{E_k}$
with:
:$E_1, E_2, \ldots, E_n$ pairwise disjoint $\Sigma_X \otimes \Sigma_Y$-measurable sets
:$a_1, a_2, \ldots, a_n$ real numbers.
We have:
{{begin-eqn}}
{{eqn | l = f^y
| r = \paren {\sum_{k \mathop = 1}^n a_k \chi_{E_k}... | Let $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$ be [[Definition:Measurable Space|measurable spaces]].
Let $\struct {X \times Y, \Sigma_X \otimes \Sigma_Y}$ be the [[Definition:Product Measurable Space|product measurable space]] of $\struct {X, \Sigma_X}$ and $\struct {Y, \Sigma_Y}$.
Let $f : X \times Y \to \R... | Write the [[Definition:Standard Representation of Simple Function|standard representation]] of $f$ as:
:$\ds f = \sum_{k \mathop = 1}^n a_k \chi_{E_k}$
with:
:$E_1, E_2, \ldots, E_n$ [[Definition:Pairwise Disjoint|pairwise disjoint]] [[Definition:Measurable Set|$\Sigma_X \otimes \Sigma_Y$-measurable]] sets
:$a_1, a... | Horizontal Section of Simple Function is Simple Function | https://proofwiki.org/wiki/Horizontal_Section_of_Simple_Function_is_Simple_Function | https://proofwiki.org/wiki/Horizontal_Section_of_Simple_Function_is_Simple_Function | [
"Horizontal Section of Functions",
"Simple Functions"
] | [
"Definition:Measurable Space",
"Definition:Product of Measurable Spaces",
"Definition:Simple Function",
"Definition:Simple Function",
"Definition:Horizontal Section of Function"
] | [
"Definition:Standard Representation of Simple Function",
"Definition:Pairwise Disjoint",
"Definition:Measurable Set",
"Definition:Real Number",
"Horizontal Section of Linear Combination of Functions is Linear Combination of Horizontal Sections",
"Horizontal Section of Characteristic Function is Characteri... |
proofwiki-18913 | Vertical Section preserves Increasing Sequences of Functions | Let $X$ and $Y$ be sets.
Let $\sequence {f_n}_{n \mathop \in \N}$ be an increasing sequence of real-valued functions with $f_i : X \times Y \to \overline \R$ for each $i$.
Let $x \in X$.
Then the sequence $\sequence {\paren {f_n}_x}_{n \mathop \in \N}$ is increasing, where $\paren {f_n}_x$ denotes the $x$-vertical se... | Since $\sequence {f_n}_{n \mathop \in \N}$ is an increasing sequence of real-valued functions, we have:
:$\map {f_i} {x, y} \le \map {f_j} {x, y}$ for all $i, j$ with $i \le j$.
for all $\tuple {x, y} \in X \times Y$.
In particular, for fixed $x \in X$, we have:
:$\map {f_i} {x, y} \le \map {f_j} {x, y}$ for all $i,... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $\sequence {f_n}_{n \mathop \in \N}$ be an [[Definition:Increasing Sequence of Real-Valued Functions|increasing sequence of real-valued functions]] with $f_i : X \times Y \to \overline \R$ for each $i$.
Let $x \in X$.
Then the [[Definition:Sequence|sequence]] $\sequ... | Since $\sequence {f_n}_{n \mathop \in \N}$ is an [[Definition:Increasing Sequence of Real-Valued Functions|increasing sequence of real-valued functions]], we have:
:$\map {f_i} {x, y} \le \map {f_j} {x, y}$ for all $i, j$ with $i \le j$.
for all $\tuple {x, y} \in X \times Y$.
In particular, for fixed $x \in X$, w... | Vertical Section preserves Increasing Sequences of Functions | https://proofwiki.org/wiki/Vertical_Section_preserves_Increasing_Sequences_of_Functions | https://proofwiki.org/wiki/Vertical_Section_preserves_Increasing_Sequences_of_Functions | [
"Vertical Section of Functions"
] | [
"Definition:Set",
"Definition:Increasing Sequence of Real-Valued Functions",
"Definition:Sequence",
"Definition:Increasing Sequence of Real-Valued Functions",
"Definition:Vertical Section of Function"
] | [
"Definition:Increasing Sequence of Real-Valued Functions",
"Definition:Vertical Section of Function",
"Definition:Increasing Sequence of Real-Valued Functions",
"Category:Vertical Section of Functions"
] |
proofwiki-18914 | Vertical Section preserves Pointwise Limits of Sequences of Functions | Let $X$ and $Y$ be sets.
Let $f : X \times Y \to \overline \R$ be a function.
Let $\sequence {f_n}_{n \mathop \in \N}$ be a sequence of functions converging pointwise to $f$.
Let $x \in X$.
Then:
:$\paren {f_n}_x \to f_x$
pointwise, where:
:$\paren {f_n}_x$ denotes the $x$-vertical section of $f_n$
:$f_x$ denotes ... | From the definition of pointwise convergence, we have:
:$\ds \map f {x, y} = \lim_{n \mathop \to \infty} \map {f_n} {x, y}$
for each $x \in X$ and $y \in Y$.
Fix $x \in X$.
From the definition of the $x$-vertical section, we have:
:$\map {f_n} {x, y} = \map {\paren {f_n}_x} y$
and:
:$\map f {x, y} = \map {f_x} y$
S... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $f : X \times Y \to \overline \R$ be a [[Definition:Extended Real-Valued Function|function]].
Let $\sequence {f_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Extended Real-Valued Function|functions]] [[Definition:Pointwise Convergence|con... | From the definition of [[Definition:Pointwise Convergence|pointwise convergence]], we have:
:$\ds \map f {x, y} = \lim_{n \mathop \to \infty} \map {f_n} {x, y}$
for each $x \in X$ and $y \in Y$.
Fix $x \in X$.
From the definition of the [[Definition:Vertical Section of Function|$x$-vertical section]], we have:
... | Vertical Section preserves Pointwise Limits of Sequences of Functions | https://proofwiki.org/wiki/Vertical_Section_preserves_Pointwise_Limits_of_Sequences_of_Functions | https://proofwiki.org/wiki/Vertical_Section_preserves_Pointwise_Limits_of_Sequences_of_Functions | [
"Vertical Section of Functions"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Sequence",
"Definition:Extended Real-Valued Function",
"Definition:Pointwise Convergence",
"Definition:Pointwise Convergence",
"Definition:Vertical Section of Function",
"Definition:Vertical Section of Function"
] | [
"Definition:Pointwise Convergence",
"Definition:Vertical Section of Function",
"Definition:Pointwise Convergence",
"Category:Vertical Section of Functions"
] |
proofwiki-18915 | Almost All Vertical Sections of Integrable Function are Integrable | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be $\sigma$-finite measure spaces.
Let $\struct {X \times Y, \Sigma_X \otimes \Sigma_Y, \mu \times \nu}$ be the product measure space of $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$.
Let $f: X \times Y \to \overline \R_{\ge 0}$ be a $\mu... | From Vertical Section of Measurable Function is Measurable, we have:
:$f_x$ is $\Sigma_Y$-measurable for each $x \in X$.
From Function Measurable iff Positive and Negative Parts Measurable, we have:
:$\paren {f_x}^+$ is $\Sigma_Y$-measurable for each $x \in X$
and:
:$\paren {f_x}^-$ is $\Sigma_Y$-measurable for each ... | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be [[Definition:Sigma-Finite Measure Space|$\sigma$-finite measure spaces]].
Let $\struct {X \times Y, \Sigma_X \otimes \Sigma_Y, \mu \times \nu}$ be the [[Definition:Product Measure Space|product measure space]] of $\struct {X, \Sigma_X, \mu}$ and $\st... | From [[Vertical Section of Measurable Function is Measurable]], we have:
:$f_x$ is [[Definition:Measurable Function|$\Sigma_Y$-measurable]] for each $x \in X$.
From [[Function Measurable iff Positive and Negative Parts Measurable]], we have:
:$\paren {f_x}^+$ is [[Definition:Measurable Function|$\Sigma_Y$-measurab... | Almost All Vertical Sections of Integrable Function are Integrable | https://proofwiki.org/wiki/Almost_All_Vertical_Sections_of_Integrable_Function_are_Integrable | https://proofwiki.org/wiki/Almost_All_Vertical_Sections_of_Integrable_Function_are_Integrable | [
"Vertical Section of Functions"
] | [
"Definition:Sigma-Finite Measure Space",
"Definition:Product Measure Space",
"Definition:Integrable Function/Measure Space",
"Definition:Integrable Function/Measure Space",
"Definition:Almost All",
"Definition:Vertical Section of Function"
] | [
"Vertical Section of Measurable Function is Measurable",
"Definition:Measurable Function",
"Function Measurable iff Positive and Negative Parts Measurable",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Extended Real-Valued Function",
"Integral of Vertical Section of Me... |
proofwiki-18916 | Almost All Horizontal Sections of Integrable Function are Integrable | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be $\sigma$-finite measure spaces.
Let $\struct {X \times Y, \Sigma_X \otimes \Sigma_Y, \mu \times \nu}$ be the product measure space of $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$.
Let $f: X \times Y \to \overline \R_{\ge 0}$ be a $\mu... | From Horizontal Section of Measurable Function is Measurable, we have:
:$f^y$ is $\Sigma_X$-measurable for each $y \in Y$.
From Function Measurable iff Positive and Negative Parts Measurable, we have:
:$\paren {f^y}^+$ is $\Sigma_X$-measurable for each $y \in Y$
and:
:$\paren {f^y}^-$ is $\Sigma_X$-measurable for eac... | Let $\struct {X, \Sigma_X, \mu}$ and $\struct {Y, \Sigma_Y, \nu}$ be [[Definition:Sigma-Finite Measure Space|$\sigma$-finite measure spaces]].
Let $\struct {X \times Y, \Sigma_X \otimes \Sigma_Y, \mu \times \nu}$ be the [[Definition:Product Measure Space|product measure space]] of $\struct {X, \Sigma_X, \mu}$ and $\st... | From [[Horizontal Section of Measurable Function is Measurable]], we have:
:$f^y$ is [[Definition:Measurable Function|$\Sigma_X$-measurable]] for each $y \in Y$.
From [[Function Measurable iff Positive and Negative Parts Measurable]], we have:
:$\paren {f^y}^+$ is [[Definition:Measurable Function|$\Sigma_X$-measur... | Almost All Horizontal Sections of Integrable Function are Integrable | https://proofwiki.org/wiki/Almost_All_Horizontal_Sections_of_Integrable_Function_are_Integrable | https://proofwiki.org/wiki/Almost_All_Horizontal_Sections_of_Integrable_Function_are_Integrable | [
"Horizontal Section of Functions"
] | [
"Definition:Sigma-Finite Measure Space",
"Definition:Product Measure Space",
"Definition:Integrable Function/Measure Space",
"Definition:Integrable Function/Measure Space",
"Definition:Almost All",
"Definition:Horizontal Section of Function"
] | [
"Horizontal Section of Measurable Function is Measurable",
"Definition:Measurable Function",
"Function Measurable iff Positive and Negative Parts Measurable",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Extended Real-Valued Function",
"Integral of Horizontal Section o... |
proofwiki-18917 | Gamma Function of 4 | :$\map \Gamma 4 = 6$ | {{begin-eqn}}
{{eqn | l = \map \Gamma 4
| r = \map \Gamma {3 + 1}
| c =
}}
{{eqn | r = 3 \map \Gamma 3
| c = Gamma Difference Equation
}}
{{eqn | r = 3 \times 2
| c = Gamma Function of 3
}}
{{end-eqn}}
{{qed}} | :$\map \Gamma 4 = 6$ | {{begin-eqn}}
{{eqn | l = \map \Gamma 4
| r = \map \Gamma {3 + 1}
| c =
}}
{{eqn | r = 3 \map \Gamma 3
| c = [[Gamma Difference Equation]]
}}
{{eqn | r = 3 \times 2
| c = [[Gamma Function of 3]]
}}
{{end-eqn}}
{{qed}} | Gamma Function of 4 | https://proofwiki.org/wiki/Gamma_Function_of_4 | https://proofwiki.org/wiki/Gamma_Function_of_4 | [
"Examples of Gamma Function Values"
] | [] | [
"Gamma Difference Equation",
"Gamma Function of 3"
] |
proofwiki-18918 | Characteristic Function of Disjoint Union | Let $X$ be a set.
Let $\sequence {D_n}_{n \mathop \in \N}$ be a sequence of pairwise disjoint subsets of $X$.
Let:
:$\ds D = \bigcup_{n \mathop = 1}^\infty D_n$
Then:
:$\ds \chi_D = \sum_{n \mathop = 1}^\infty \chi_{D_n}$
where:
:$\chi_D$ is the characteristic function of $D$
:$\chi_{D_n}$ is the characteristic fun... | We aim to show that:
:$\ds \sum_{n \mathop = 1}^\infty \map {\chi_{D_n} } x = \begin{cases}1 & x \in D \\ 0 & x \in X \setminus D\end{cases}$
at which point we will have the demand from the definition of a characteristic function.
Let $x \in D$.
Then:
:$\ds x \in \bigcup_{n \mathop = 1}^\infty D_n$
From the definit... | Let $X$ be a [[Definition:Set|set]].
Let $\sequence {D_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Pairwise Disjoint|pairwise disjoint]] subsets of $X$.
Let:
:$\ds D = \bigcup_{n \mathop = 1}^\infty D_n$
Then:
:$\ds \chi_D = \sum_{n \mathop = 1}^\infty \chi_{D_n}$
where:
:$\... | We aim to show that:
:$\ds \sum_{n \mathop = 1}^\infty \map {\chi_{D_n} } x = \begin{cases}1 & x \in D \\ 0 & x \in X \setminus D\end{cases}$
at which point we will have the demand from the definition of a [[Definition:Characteristic Function (Set Theory)|characteristic function]].
Let $x \in D$.
Then:
:$\ds x... | Characteristic Function of Disjoint Union | https://proofwiki.org/wiki/Characteristic_Function_of_Disjoint_Union | https://proofwiki.org/wiki/Characteristic_Function_of_Disjoint_Union | [
"Characteristic Functions",
"Set Union",
"Characteristic Function of Disjoint Union"
] | [
"Definition:Set",
"Definition:Sequence",
"Definition:Pairwise Disjoint",
"Definition:Characteristic Function (Set Theory)",
"Definition:Characteristic Function (Set Theory)"
] | [
"Definition:Characteristic Function (Set Theory)",
"Definition:Set Union",
"Definition:Pairwise Disjoint",
"Definition:Characteristic Function (Set Theory)",
"Definition:Set Difference",
"Definition:Set Union",
"Definition:Characteristic Function (Set Theory)",
"Category:Characteristic Functions",
"... |
proofwiki-18919 | Integral of Positive Measurable Function over Disjoint Union | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f : X \to \overline \R$ be a positive $\Sigma$-measurable function.
Let $\sequence {D_n}_{n \mathop \in \N}$ be a sequence of pairwise disjoint $\Sigma$-measurable sets.
Let:
:$\ds D = \bigcup_{n \mathop = 1}^\infty D_n$
Then:
:$\ds \int_D f \rd \mu = \sum_{n \m... | We have:
{{begin-eqn}}
{{eqn | l = \int_D f \rd \mu
| r = \int \paren {f \times \chi_D} \rd \mu
| c = {{Defof|Integral of Positive Measurable Function over Measurable Set}}
}}
{{eqn | r = \int \paren {f \times \paren {\sum_{n \mathop = 1}^\infty \chi_{D_n} } } \rd \mu
| c = Characteristic Function of Disjoint U... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f : X \to \overline \R$ be a [[Definition:Positive Measurable Function|positive $\Sigma$-measurable function]].
Let $\sequence {D_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Pairwise Disjoint|pairwise... | We have:
{{begin-eqn}}
{{eqn | l = \int_D f \rd \mu
| r = \int \paren {f \times \chi_D} \rd \mu
| c = {{Defof|Integral of Positive Measurable Function over Measurable Set}}
}}
{{eqn | r = \int \paren {f \times \paren {\sum_{n \mathop = 1}^\infty \chi_{D_n} } } \rd \mu
| c = [[Characteristic Function of Disjoin... | Integral of Positive Measurable Function over Disjoint Union | https://proofwiki.org/wiki/Integral_of_Positive_Measurable_Function_over_Disjoint_Union | https://proofwiki.org/wiki/Integral_of_Positive_Measurable_Function_over_Disjoint_Union | [
"Integral of Positive Measurable Function over Measurable Set"
] | [
"Definition:Measure Space",
"Definition:Measurable Function/Positive",
"Definition:Sequence",
"Definition:Pairwise Disjoint",
"Definition:Measurable Set"
] | [
"Characteristic Function of Disjoint Union",
"Integral of Series of Positive Measurable Functions",
"Category:Integral of Positive Measurable Function over Measurable Set"
] |
proofwiki-18920 | Integral of Positive Measurable Function over Measurable Set is Well-Defined | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $A \in \Sigma$.
Let $f : X \to \overline \R$ be a positive $\Sigma$-measurable function.
Then the $\mu$-integral of $f$ over $A$ defined by:
:$\ds \int_A f \rd \mu = \int \paren {\chi_A \cdot f} \rd \mu$
is well-defined. | We simply need to show that:
:$\chi_A \cdot f$ is a positive $\Sigma$-measurable function.
For $x \in A$, we have:
{{begin-eqn}}
{{eqn | l = \map {\paren {\chi_A \cdot f} } x
| r = \map {\chi_A} x \map f x
| c = {{Defof|Pointwise Multiplication}}
}}
{{eqn | r = \map f x
| c = {{Defof|Characteristic Function of ... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $A \in \Sigma$.
Let $f : X \to \overline \R$ be a [[Definition:Positive Measurable Function|positive $\Sigma$-measurable function]].
Then the [[Definition:Integral of Positive Measurable Function over Measurable Set|$\mu$-integral ... | We simply need to show that:
:$\chi_A \cdot f$ is a [[Definition:Positive Measurable Function|positive $\Sigma$-measurable function]].
For $x \in A$, we have:
{{begin-eqn}}
{{eqn | l = \map {\paren {\chi_A \cdot f} } x
| r = \map {\chi_A} x \map f x
| c = {{Defof|Pointwise Multiplication}}
}}
{{eqn | r = \map ... | Integral of Positive Measurable Function over Measurable Set is Well-Defined | https://proofwiki.org/wiki/Integral_of_Positive_Measurable_Function_over_Measurable_Set_is_Well-Defined | https://proofwiki.org/wiki/Integral_of_Positive_Measurable_Function_over_Measurable_Set_is_Well-Defined | [
"Integral of Positive Measurable Function over Measurable Set"
] | [
"Definition:Measure Space",
"Definition:Measurable Function/Positive",
"Definition:Integral of Positive Measurable Function over Measurable Set"
] | [
"Definition:Measurable Function/Positive",
"Definition:Measurable Function",
"Characteristic Function Measurable iff Set Measurable",
"Definition:Measurable Function",
"Pointwise Product of Measurable Functions is Measurable",
"Definition:Measurable Function",
"Definition:Measurable Function/Positive",
... |
proofwiki-18921 | Measurable Function is Integrable iff A.E. Equal to Real-Valued Integrable Function | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f : X \to \overline \R$ be a $\Sigma$-measurable function.
Then $f$ is $\mu$-integrable {{iff}}:
:there exists a $\mu$-integrable function $g : X \to \R$ such that $g = f$ $\mu$-almost everywhere. | === Sufficient Condition ===
Suppose that:
:there exists a $\mu$-integrable function $g : X \to \R$ such that $g = f$ $\mu$-almost everywhere.
Then, from A.E. Equal Positive Measurable Functions have Equal Integrals: Corollary 1, we have:
:$f$ is $\mu$-integrable.
{{qed|lemma}} | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f : X \to \overline \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
Then $f$ is [[Definition:Measure-Integrable Function|$\mu$-integrable]] {{iff}}:
:there exists a [[Definition:Measure-Integrable Function... | === Sufficient Condition ===
Suppose that:
:there exists a [[Definition:Measure-Integrable Function|$\mu$-integrable function]] $g : X \to \R$ such that $g = f$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]].
Then, from [[A.E. Equal Positive Measurable Functions have Equal Integrals/Corollary 1|A.E. Equal... | Measurable Function is Integrable iff A.E. Equal to Real-Valued Integrable Function | https://proofwiki.org/wiki/Measurable_Function_is_Integrable_iff_A.E._Equal_to_Real-Valued_Integrable_Function | https://proofwiki.org/wiki/Measurable_Function_is_Integrable_iff_A.E._Equal_to_Real-Valued_Integrable_Function | [
"Measure Theory"
] | [
"Definition:Measure Space",
"Definition:Measurable Function",
"Definition:Integrable Function/Measure Space",
"Definition:Integrable Function/Measure Space",
"Definition:Almost Everywhere"
] | [
"Definition:Integrable Function/Measure Space",
"Definition:Almost Everywhere",
"A.E. Equal Positive Measurable Functions have Equal Integrals/Corollary 1",
"Definition:Integrable Function/Measure Space",
"Definition:Integrable Function/Measure Space",
"Definition:Almost Everywhere",
"A.E. Equal Positiv... |
proofwiki-18922 | Integral of Positive Measurable Function is Additive/Corollary | Let $A \in \Sigma$.
Then:
:$\ds \int_A \paren {f + g} \rd \mu = \int_A f \rd \mu + \int_A g \rd \mu$
where:
:$f + g$ is the pointwise sum of $f$ and $g$
:the integral sign denotes $\mu$-integration over $A$.
This can be summarized by saying that $\ds \int_A \cdot \rd \mu$ is additive. | We have:
{{begin-eqn}}
{{eqn | l = \int_A \paren {f + g} \rd \mu
| r = \int \paren {f + g} \times \chi_A \rd \mu
| c = {{Defof|Integral of Positive Measurable Function over Measurable Set}}
}}
{{eqn | r = \int \paren {f \times \chi_A + g \times \chi_A} \rd \mu
}}
{{eqn | r = \int \paren {f \times \chi_A} \rd \mu +... | Let $A \in \Sigma$.
Then:
:$\ds \int_A \paren {f + g} \rd \mu = \int_A f \rd \mu + \int_A g \rd \mu$
where:
:$f + g$ is the [[Definition:Pointwise Addition|pointwise sum]] of $f$ and $g$
:the [[Definition:Integral Sign|integral sign]] denotes [[Definition:Integral of Positive Measurable Function over Measurable S... | We have:
{{begin-eqn}}
{{eqn | l = \int_A \paren {f + g} \rd \mu
| r = \int \paren {f + g} \times \chi_A \rd \mu
| c = {{Defof|Integral of Positive Measurable Function over Measurable Set}}
}}
{{eqn | r = \int \paren {f \times \chi_A + g \times \chi_A} \rd \mu
}}
{{eqn | r = \int \paren {f \times \chi_A} \rd \mu ... | Integral of Positive Measurable Function is Additive/Corollary | https://proofwiki.org/wiki/Integral_of_Positive_Measurable_Function_is_Additive/Corollary | https://proofwiki.org/wiki/Integral_of_Positive_Measurable_Function_is_Additive/Corollary | [
"Integral of Positive Measurable Function is Additive",
"Integral of Positive Measurable Function over Measurable Set"
] | [
"Definition:Pointwise Addition",
"Definition:Integral Sign",
"Definition:Integral of Positive Measurable Function over Measurable Set",
"Definition:Additive Function (Algebra)"
] | [
"Integral of Positive Measurable Function is Additive",
"Category:Integral of Positive Measurable Function is Additive",
"Category:Integral of Positive Measurable Function over Measurable Set"
] |
proofwiki-18923 | Linear Combination of Signed Measures is Signed Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ and $\nu$ be signed measures on $\struct {X, \Sigma}$.
Let $\alpha, \beta \in \overline \R$.
Suppose that the sum:
:$\alpha \map \mu A + \beta \map \nu A$
is well-defined for each $A \in \Sigma$.
Then:
:$\xi = \alpha \mu + \beta \nu$ is a signed measure. | We verify both of the conditions for a signed measure. | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ and $\nu$ be [[Definition:Signed Measure|signed measures]] on $\struct {X, \Sigma}$.
Let $\alpha, \beta \in \overline \R$.
Suppose that the [[Definition:Extended Real Addition|sum]]:
:$\alpha \map \mu A + \beta \map \nu A$
... | We verify both of the conditions for a [[Definition:Signed Measure|signed measure]]. | Linear Combination of Signed Measures is Signed Measure | https://proofwiki.org/wiki/Linear_Combination_of_Signed_Measures_is_Signed_Measure | https://proofwiki.org/wiki/Linear_Combination_of_Signed_Measures_is_Signed_Measure | [
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Extended Real Addition",
"Definition:Signed Measure"
] | [
"Definition:Signed Measure",
"Definition:Signed Measure",
"Definition:Signed Measure"
] |
proofwiki-18924 | Absolute Value of Signed Measure Bounded Above by Variation | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\size \mu$ be the variation of $\mu$.
Then:
:$\size {\map \mu A} \le \map {\size \mu} A$
for each $A \in \Sigma$. | Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.
Then:
:$\mu = \mu^+ - \mu^-$
and:
:$\size \mu = \mu^+ + \mu^-$
We have:
{{begin-eqn}}
{{eqn | l = \size {\map \mu A}
| r = \size {\map {\mu^+} A - \map {\mu^-} A}
}}
{{eqn | o = \le
| r = \size {\map {\mu^+} A} + \size {\map {\mu^-} A}
| c = Tria... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $\size \mu$ be the [[Definition:Variation of Signed Measure|variation]] of $\mu$.
Then:
:$\size {\map \mu A} \le \map {\size \mu} A$
for ea... | Let $\tuple {\mu^+, \mu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$.
Then:
:$\mu = \mu^+ - \mu^-$
and:
:$\size \mu = \mu^+ + \mu^-$
We have:
{{begin-eqn}}
{{eqn | l = \size {\map \mu A}
| r = \size {\map {\mu^+} A - \map {\mu^-} A}
}}
{{eqn | o = \le
| r = \size {\map {\mu^+... | Absolute Value of Signed Measure Bounded Above by Variation | https://proofwiki.org/wiki/Absolute_Value_of_Signed_Measure_Bounded_Above_by_Variation | https://proofwiki.org/wiki/Absolute_Value_of_Signed_Measure_Bounded_Above_by_Variation | [
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Variation/Signed Measure"
] | [
"Definition:Jordan Decomposition",
"Triangle Inequality"
] |
proofwiki-18925 | Decomposition of Complex Measure into Finite Signed Measures | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.
Then there exists unique finite signed measures $\mu_R$ and $\mu_I$ such that:
:$\mu = \mu_R + i \mu_I$ | === Existence ===
For each $A \in \Sigma$ define the function $\mu_R : X \to \R$ by:
:$\map {\mu_R} A = \map \Re {\map \mu A}$
Similarly, for each $A \in \Sigma$ define the function $\mu_I : X \to \R$ by:
:$\map {\mu_I} A = \map \Im {\map \mu A}$
Clearly we have:
{{begin-eqn}}
{{eqn | l = \map \mu A
| r = \map \Re... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Complex Measure|complex measure]] on $\struct {X, \Sigma}$.
Then there exists unique [[Definition:Finite Signed Measure|finite signed measures]] $\mu_R$ and $\mu_I$ such that:
:$\mu = \mu_R + i \mu_I$ | === Existence ===
For each $A \in \Sigma$ define the [[Definition:Real-Valued Function|function]] $\mu_R : X \to \R$ by:
:$\map {\mu_R} A = \map \Re {\map \mu A}$
Similarly, for each $A \in \Sigma$ define the [[Definition:Real-Valued Function|function]] $\mu_I : X \to \R$ by:
:$\map {\mu_I} A = \map \Im {\map \mu... | Decomposition of Complex Measure into Finite Signed Measures | https://proofwiki.org/wiki/Decomposition_of_Complex_Measure_into_Finite_Signed_Measures | https://proofwiki.org/wiki/Decomposition_of_Complex_Measure_into_Finite_Signed_Measures | [
"Complex Measures"
] | [
"Definition:Measurable Space",
"Definition:Complex Measure",
"Definition:Finite Measure/Signed Measure"
] | [
"Definition:Real-Valued Function",
"Definition:Real-Valued Function",
"Definition:Finite Measure/Signed Measure",
"Definition:Signed Measure",
"Definition:Sequence",
"Definition:Pairwise Disjoint",
"Definition:Measurable Set",
"Definition:Countably Additive Function",
"Definition:Countably Additive ... |
proofwiki-18926 | Measurable Function Zero A.E. iff Absolute Value has Zero Integral/Corollary | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f: X \to \overline \R$ be a non-negative integrable function.
Let $A, B \in \Sigma$ have $A \subseteq B$.
Then:
:$\ds \int_A f \rd \mu = \int_B f \rd \mu$
{{iff}}:
:$f \times \chi_{B \setminus A} = 0$ $\mu$-almost everywhere. | We can write:
:$B = A \cup \paren {B \setminus A}$
From Integral of Positive Measurable Function over Disjoint Union, we have:
:$\ds \int_B f \rd \mu = \int_A f \rd \mu + \int_{B \setminus A} f \rd \mu$
Since:
:$\ds \int_B f \rd \mu = \int_A f \rd \mu$
we get:
:$\ds \int_{B \setminus A} f \rd \mu = 0$
From the defi... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f: X \to \overline \R$ be a non-negative [[Definition:Integrable Function|integrable function]].
Let $A, B \in \Sigma$ have $A \subseteq B$.
Then:
:$\ds \int_A f \rd \mu = \int_B f \rd \mu$
{{iff}}:
:$f \times \chi_{B \setmin... | We can write:
:$B = A \cup \paren {B \setminus A}$
From [[Integral of Positive Measurable Function over Disjoint Union]], we have:
:$\ds \int_B f \rd \mu = \int_A f \rd \mu + \int_{B \setminus A} f \rd \mu$
Since:
:$\ds \int_B f \rd \mu = \int_A f \rd \mu$
we get:
:$\ds \int_{B \setminus A} f \rd \mu = 0$
F... | Measurable Function Zero A.E. iff Absolute Value has Zero Integral/Corollary | https://proofwiki.org/wiki/Measurable_Function_Zero_A.E._iff_Absolute_Value_has_Zero_Integral/Corollary | https://proofwiki.org/wiki/Measurable_Function_Zero_A.E._iff_Absolute_Value_has_Zero_Integral/Corollary | [
"Measurable Function Zero A.E. iff Absolute Value has Zero Integral"
] | [
"Definition:Measure Space",
"Definition:Integrable Function",
"Definition:Almost Everywhere"
] | [
"Integral of Positive Measurable Function over Disjoint Union",
"Definition:Integral of Positive Measurable Function over Measurable Set",
"Measurable Function Zero A.E. iff Absolute Value has Zero Integral",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Category:Measurable Function Zer... |
proofwiki-18927 | Jordan Decomposition of Finite Signed Measure | Let $\struct {X, \Sigma}$ be measurable.
Let $\mu$ be a finite signed measure on $\struct {X, \Sigma}$.
Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.
Then $\mu^+$ and $\mu^-$ are finite measures. | From the definition of Jordan decomposition, we have:
:$\mu = \mu^+ - \mu^-$
with at least one of $\mu^+$ and $\mu^-$ finite.
From the definition of a finite signed measure, we have:
:$\cmod {\map \mu X} < \infty$
We show that:
:if exactly one of $\mu^+$ and $\mu^-$ is finite, then $\mu$ is not a finite signed measu... | Let $\struct {X, \Sigma}$ be [[Definition:Measurable Space|measurable]].
Let $\mu$ be a [[Definition:Finite Signed Measure|finite signed measure]] on $\struct {X, \Sigma}$.
Let $\tuple {\mu^+, \mu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$.
Then $\mu^+$ and $\mu^-$ are [[Definitio... | From the definition of [[Definition:Jordan Decomposition|Jordan decomposition]], we have:
:$\mu = \mu^+ - \mu^-$
with at least one of $\mu^+$ and $\mu^-$ [[Definition:Finite Measure|finite]].
From the definition of a [[Definition:Finite Signed Measure|finite signed measure]], we have:
:$\cmod {\map \mu X} < \infty... | Jordan Decomposition of Finite Signed Measure | https://proofwiki.org/wiki/Jordan_Decomposition_of_Finite_Signed_Measure | https://proofwiki.org/wiki/Jordan_Decomposition_of_Finite_Signed_Measure | [
"Signed Measures",
"Finite Signed Measures",
"Finite Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Finite Measure/Signed Measure",
"Definition:Jordan Decomposition",
"Definition:Finite Measure"
] | [
"Definition:Jordan Decomposition",
"Definition:Finite Measure",
"Definition:Finite Measure/Signed Measure",
"Definition:Finite Measure",
"Definition:Finite Measure/Signed Measure",
"Definition:Finite Measure",
"Definition:Finite Measure",
"Definition:Finite Measure",
"Definition:Finite Measure",
"... |
proofwiki-18928 | Integral of Bounded Measurable Function with respect to Finite Signed Measure is Well-Defined | Let $\struct {X, \Sigma}$ be a measurable space.
Let $f : X \to \R$ be a bounded $\Sigma$-measurable function.
Let $\mu$ be a finite signed measure on $\struct {X, \Sigma}$.
Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.
Then the $\mu$-integral of $f$ defined by:
:$\ds \int f \rd \mu = \int f \rd ... | We show that $f$ is $\mu^+$-integrable and $\mu^-$-integrable.
We will then have:
:$\ds -\infty < \int f \rd \mu^+ < \infty$
and:
:$\ds -\infty < \int f \rd \mu^- < \infty$
So that:
:$\ds \int f \rd \mu^+ - \int f \rd \mu^-$
is well-defined.
Since $f$ is bounded, there exists $M > 0$ such that:
:$\size {\map f x} ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $f : X \to \R$ be a [[Definition:Bounded Real-Valued Function|bounded]] [[Definition:Measurable Function|$\Sigma$-measurable function]].
Let $\mu$ be a [[Definition:Finite Signed Measure|finite signed measure]] on $\struct {X, \Sigma... | We show that $f$ is [[Definition:Measure-Integrable Function|$\mu^+$-integrable]] and [[Definition:Measure-Integrable Function|$\mu^-$-integrable]].
We will then have:
:$\ds -\infty < \int f \rd \mu^+ < \infty$
and:
:$\ds -\infty < \int f \rd \mu^- < \infty$
So that:
:$\ds \int f \rd \mu^+ - \int f \rd \mu^-$ ... | Integral of Bounded Measurable Function with respect to Finite Signed Measure is Well-Defined | https://proofwiki.org/wiki/Integral_of_Bounded_Measurable_Function_with_respect_to_Finite_Signed_Measure_is_Well-Defined | https://proofwiki.org/wiki/Integral_of_Bounded_Measurable_Function_with_respect_to_Finite_Signed_Measure_is_Well-Defined | [
"Integral of Bounded Measurable Function with respect to Finite Signed Measure"
] | [
"Definition:Measurable Space",
"Definition:Bounded Mapping/Real-Valued",
"Definition:Measurable Function",
"Definition:Finite Measure/Signed Measure",
"Definition:Jordan Decomposition",
"Definition:Integral of Bounded Measurable Function with respect to Finite Signed Measure",
"Definition:Well-Defined"
... | [
"Definition:Integrable Function/Measure Space",
"Definition:Integrable Function/Measure Space",
"Definition:Well-Defined",
"Definition:Bounded Mapping/Real-Valued",
"Jordan Decomposition of Finite Signed Measure",
"Definition:Finite Measure",
"Measure is Monotone",
"Integral of Positive Measurable Fun... |
proofwiki-18929 | Positive Part of Vertical Section of Function is Vertical Section of Positive Part | Let $X$ and $Y$ be sets.
Let $f : X \times Y \to \overline \R$ be a function.
Let $x \in X$.
Then:
:$\paren {f_x}^+ = \paren {f^+}_x$
where:
:$f_x$ denotes the $x$-vertical function of $f$
:$f^+$ denotes the positive part of $f$. | Fix $x \in X$.
Then, we have, for each $y \in Y$:
{{begin-eqn}}
{{eqn | l = \map {\paren {f^+}_x} y
| r = \map {f^+} {x, y}
}}
{{eqn | r = \max \set {0, \map f {x, y} }
| c = {{Defof|Positive Part}}
}}
{{eqn | r = \max \set {0, \map {f_x} y}
| c = {{Defof|Vertical Section of Function}}
}}
{{eqn | r = \map {\pare... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $f : X \times Y \to \overline \R$ be a [[Definition:Extended Real-Valued Function|function]].
Let $x \in X$.
Then:
:$\paren {f_x}^+ = \paren {f^+}_x$
where:
:$f_x$ denotes the [[Definition:Vertical Section of Function|$x$-vertical function]] of $f$
:$f^+$ denote... | Fix $x \in X$.
Then, we have, for each $y \in Y$:
{{begin-eqn}}
{{eqn | l = \map {\paren {f^+}_x} y
| r = \map {f^+} {x, y}
}}
{{eqn | r = \max \set {0, \map f {x, y} }
| c = {{Defof|Positive Part}}
}}
{{eqn | r = \max \set {0, \map {f_x} y}
| c = {{Defof|Vertical Section of Function}}
}}
{{eqn | r = \map {\pa... | Positive Part of Vertical Section of Function is Vertical Section of Positive Part | https://proofwiki.org/wiki/Positive_Part_of_Vertical_Section_of_Function_is_Vertical_Section_of_Positive_Part | https://proofwiki.org/wiki/Positive_Part_of_Vertical_Section_of_Function_is_Vertical_Section_of_Positive_Part | [
"Positive Parts",
"Vertical Section of Functions",
"Positive Parts"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Vertical Section of Function",
"Definition:Positive Part"
] | [
"Category:Vertical Section of Functions",
"Category:Positive Parts"
] |
proofwiki-18930 | Negative Part of Vertical Section of Function is Vertical Section of Negative Part | Let $X$ and $Y$ be sets.
Let $f : X \times Y \to \overline \R$ be a function.
Let $x \in X$.
Then:
:$\paren {f_x}^- = \paren {f^-}_x$
where:
:$f_x$ denotes the $x$-vertical function of $f$
:$f^-$ denotes the negative part of $f$. | Fix $x \in X$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \map {\paren {f^-}_x} y
| r = \map {f^-} {x, y}
}}
{{eqn | r = -\min \set {0, \map f {x, y} }
| c = {{Defof|Negative Part}}
}}
{{eqn | r = -\min \set {0, \map {f_x} y}
| c = {{Defof|Vertical Section of Function}}
}}
{{eqn | r = \map {\paren {f_x}^-} y
| c... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $f : X \times Y \to \overline \R$ be a [[Definition:Extended Real-Valued Function|function]].
Let $x \in X$.
Then:
:$\paren {f_x}^- = \paren {f^-}_x$
where:
:$f_x$ denotes the [[Definition:Vertical Section of Function|$x$-vertical function]] of $f$
:$f^-$ denote... | Fix $x \in X$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \map {\paren {f^-}_x} y
| r = \map {f^-} {x, y}
}}
{{eqn | r = -\min \set {0, \map f {x, y} }
| c = {{Defof|Negative Part}}
}}
{{eqn | r = -\min \set {0, \map {f_x} y}
| c = {{Defof|Vertical Section of Function}}
}}
{{eqn | r = \map {\paren {f_x}^-} y
|... | Negative Part of Vertical Section of Function is Vertical Section of Negative Part | https://proofwiki.org/wiki/Negative_Part_of_Vertical_Section_of_Function_is_Vertical_Section_of_Negative_Part | https://proofwiki.org/wiki/Negative_Part_of_Vertical_Section_of_Function_is_Vertical_Section_of_Negative_Part | [
"Negative Parts",
"Vertical Section of Functions",
"Negative Parts"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Vertical Section of Function",
"Definition:Negative Part"
] | [
"Category:Vertical Section of Functions",
"Category:Negative Parts"
] |
proofwiki-18931 | Positive Part of Horizontal Section of Function is Horizontal Section of Positive Part | Let $X$ and $Y$ be sets.
Let $f : X \times Y \to \overline \R$ be a function.
Let $y \in Y$.
Then:
:$\paren {f^y}^+ = \paren {f^+}^y$
where:
:$f^y$ denotes the $y$-horizontal function of $f$
:$f^+$ denotes the positive part of $f$. | Fix $y \in Y$.
Then, we have, for each $x \in X$:
{{begin-eqn}}
{{eqn | l = \map {\paren {f^+}^y} x
| r = \map {f^+} {x, y}
}}
{{eqn | r = \max \set {0, \map f {x, y} }
| c = {{Defof|Positive Part}}
}}
{{eqn | r = \max \set {0, \map {f^y} x}
| c = {{Defof|Horizontal Section of Function}}
}}
{{eqn | r = \map {\pa... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $f : X \times Y \to \overline \R$ be a [[Definition:Extended Real-Valued Function|function]].
Let $y \in Y$.
Then:
:$\paren {f^y}^+ = \paren {f^+}^y$
where:
:$f^y$ denotes the [[Definition:Horizontal Section of Function|$y$-horizontal function]] of $f$
:$f^+$ de... | Fix $y \in Y$.
Then, we have, for each $x \in X$:
{{begin-eqn}}
{{eqn | l = \map {\paren {f^+}^y} x
| r = \map {f^+} {x, y}
}}
{{eqn | r = \max \set {0, \map f {x, y} }
| c = {{Defof|Positive Part}}
}}
{{eqn | r = \max \set {0, \map {f^y} x}
| c = {{Defof|Horizontal Section of Function}}
}}
{{eqn | r = \map {\... | Positive Part of Horizontal Section of Function is Horizontal Section of Positive Part | https://proofwiki.org/wiki/Positive_Part_of_Horizontal_Section_of_Function_is_Horizontal_Section_of_Positive_Part | https://proofwiki.org/wiki/Positive_Part_of_Horizontal_Section_of_Function_is_Horizontal_Section_of_Positive_Part | [
"Positive Parts",
"Horizontal Section of Functions",
"Positive Parts"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Horizontal Section of Function",
"Definition:Positive Part"
] | [
"Category:Horizontal Section of Functions",
"Category:Positive Parts"
] |
proofwiki-18932 | Negative Part of Horizontal Section of Function is Horizontal Section of Negative Part | Let $X$ and $Y$ be sets.
Let $f : X \times Y \to \overline \R$ be a function.
Let $y \in Y$.
Then:
:$\paren {f^y}^- = \paren {f^-}^y$
where:
:$f^y$ denotes the $y$-horizontal function of $f$
:$f^-$ denotes the negative part of $f$. | Fix $y \in Y$.
Then, we have, for each $x \in X$:
{{begin-eqn}}
{{eqn | l = \map {\paren {f^-}^y} x
| r = \map {f^-} {x, y}
}}
{{eqn | r = -\min \set {0, \map f {x, y} }
| c = {{Defof|Negative Part}}
}}
{{eqn | r = -\min \set {0, \map {f^y} x}
| c = {{Defof|Horizontal Section of Function}}
}}
{{eqn | r = \map {\... | Let $X$ and $Y$ be [[Definition:Set|sets]].
Let $f : X \times Y \to \overline \R$ be a [[Definition:Extended Real-Valued Function|function]].
Let $y \in Y$.
Then:
:$\paren {f^y}^- = \paren {f^-}^y$
where:
:$f^y$ denotes the [[Definition:Horizontal Section of Function|$y$-horizontal function]] of $f$
:$f^-$ de... | Fix $y \in Y$.
Then, we have, for each $x \in X$:
{{begin-eqn}}
{{eqn | l = \map {\paren {f^-}^y} x
| r = \map {f^-} {x, y}
}}
{{eqn | r = -\min \set {0, \map f {x, y} }
| c = {{Defof|Negative Part}}
}}
{{eqn | r = -\min \set {0, \map {f^y} x}
| c = {{Defof|Horizontal Section of Function}}
}}
{{eqn | r = \map ... | Negative Part of Horizontal Section of Function is Horizontal Section of Negative Part | https://proofwiki.org/wiki/Negative_Part_of_Horizontal_Section_of_Function_is_Horizontal_Section_of_Negative_Part | https://proofwiki.org/wiki/Negative_Part_of_Horizontal_Section_of_Function_is_Horizontal_Section_of_Negative_Part | [
"Negative Parts",
"Horizontal Section of Functions",
"Negative Parts"
] | [
"Definition:Set",
"Definition:Extended Real-Valued Function",
"Definition:Horizontal Section of Function",
"Definition:Negative Part"
] | [
"Category:Horizontal Section of Functions",
"Category:Negative Parts"
] |
proofwiki-18933 | Characteristic Function of Null Set is A.E. Equal to Zero | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $N$ be a $\mu$-null set.
Then:
:$\chi_N = 0$ $\mu$-almost everywhere.
where $\chi_N$ is the characteristic function of $N$. | Let $x \in X$ be such that:
:$\map {\chi_N} x \ne 0$
Then. since $\map {\chi_N} x \in \set {0, 1}$:
:$\map {\chi_N} x = 1$
which is equivalent to:
:$x \in N$
from the definition of a characteristic function.
So:
:if $x \in X$ is such that $\map {\chi_N} x \ne 0$, then $x \in N$.
Since $N$ is a $\mu$-null set, we ha... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $N$ be a [[Definition:Null Set|$\mu$-null set]].
Then:
:$\chi_N = 0$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]].
where $\chi_N$ is the [[Definition:Characteristic Function of Set|characteristic function]] of $N$. | Let $x \in X$ be such that:
:$\map {\chi_N} x \ne 0$
Then. since $\map {\chi_N} x \in \set {0, 1}$:
:$\map {\chi_N} x = 1$
which is equivalent to:
:$x \in N$
from the definition of a [[Definition:Characteristic Function of Set|characteristic function]].
So:
:if $x \in X$ is such that $\map {\chi_N} x \ne 0$... | Characteristic Function of Null Set is A.E. Equal to Zero | https://proofwiki.org/wiki/Characteristic_Function_of_Null_Set_is_A.E._Equal_to_Zero | https://proofwiki.org/wiki/Characteristic_Function_of_Null_Set_is_A.E._Equal_to_Zero | [
"Characteristic Functions",
"Measure Theory",
"Characteristic Function of Null Set is A.E. Equal to Zero"
] | [
"Definition:Measure Space",
"Definition:Null Set",
"Definition:Almost Everywhere",
"Definition:Characteristic Function (Set Theory)/Set"
] | [
"Definition:Characteristic Function (Set Theory)/Set",
"Definition:Null Set",
"Definition:Almost Everywhere",
"Category:Characteristic Functions",
"Category:Measure Theory",
"Category:Characteristic Function of Null Set is A.E. Equal to Zero"
] |
proofwiki-18934 | Pointwise Addition preserves A.E. Equality | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f, g, F, G : X \to \overline \R$ be functions with:
:$f = F$ $\mu$-almost everywhere
and:
:$g = G$ $\mu$-almost everywhere
and the pointwise sums $f + g$ and $F + G$ well-defined.
Then:
:$f + g = F + G$ $\mu$-almost everywhere. | Since:
:$f = F$ $\mu$-almost everywhere
there exists a $\mu$-null set $N_1 \subseteq X$ such that:
:if $x \in X$ has $\map f x \ne \map F x$ then $x \in N_1$.
Since:
:$g = G$ $\mu$-almost everywhere
there exists a $\mu$-null set $N_2 \subseteq X$ such that:
:if $x \in X$ has $\map G x \ne \map G x$ then $x \in N_2$ ... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f, g, F, G : X \to \overline \R$ be [[Definition:Extended Real-Valued Function|functions]] with:
:$f = F$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]]
and:
:$g = G$ [[Definition:Almost Everywhere|$\mu$-almost everywher... | Since:
:$f = F$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]]
there exists a [[Definition:Null Set|$\mu$-null set]] $N_1 \subseteq X$ such that:
:if $x \in X$ has $\map f x \ne \map F x$ then $x \in N_1$.
Since:
:$g = G$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]]
there exists a [[Defini... | Pointwise Addition preserves A.E. Equality | https://proofwiki.org/wiki/Pointwise_Addition_preserves_A.E._Equality | https://proofwiki.org/wiki/Pointwise_Addition_preserves_A.E._Equality | [
"Measure Theory",
"Almost-Everywhere Equality Relation",
"Almost-Everywhere Equality Relation"
] | [
"Definition:Measure Space",
"Definition:Extended Real-Valued Function",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Pointwise Addition of Extended Real-Valued Functions",
"Definition:Almost Everywhere"
] | [
"Definition:Almost Everywhere",
"Definition:Null Set",
"Definition:Almost Everywhere",
"Definition:Null Set",
"Definition:Pointwise Addition of Extended Real-Valued Functions",
"Rule of Transposition",
"Null Sets Closed under Countable Union",
"Definition:Null Set",
"Definition:Almost Everywhere",
... |
proofwiki-18935 | Lebesgue 1-Space is Subset of Tempered Distribution Space | Let $\map {L^1} \R$ be the Lebesgue $1$-space.
Let $\map {\SS'} \R$ be the tempered distribution space.
Then in the distributional sense:
:$\map {L^1} \R \subseteq \map {\SS'} \R$
That is:
:$T_f \subseteq \map {\SS'} \R$
where $f \in \map {L^1} \R$. | Let $f \in \map {L^1} \R$.
By definition of the Lebesgue space:
:$\ds \norm f_1 = \int_\R \size {\map f x} \rd x < \infty$
where $\norm {\, \cdot \,}_1$ denotes the 1-seminorm.
Let $\phi \in \map \SS \R$ be a Schwartz test function, where $\map \SS \R$ is the Schwartz space.
Let $T_f : \map \SS \R \to \R$ be a function... | Let $\map {L^1} \R$ be the [[Definition:Lebesgue Space|Lebesgue $1$-space]].
Let $\map {\SS'} \R$ be the [[Definition:Tempered Distribution Space|tempered distribution space]].
Then in the [[Definition:Tempered Distribution|distributional sense]]:
:$\map {L^1} \R \subseteq \map {\SS'} \R$
That is:
:$T_f \subseteq... | Let $f \in \map {L^1} \R$.
By definition of the [[Definition:Lebesgue Space|Lebesgue space]]:
:$\ds \norm f_1 = \int_\R \size {\map f x} \rd x < \infty$
where $\norm {\, \cdot \,}_1$ denotes the [[Definition:P-Seminorm|1-seminorm]].
Let $\phi \in \map \SS \R$ be a [[Definition:Schwartz Test Function|Schwartz test f... | Lebesgue 1-Space is Subset of Tempered Distribution Space | https://proofwiki.org/wiki/Lebesgue_1-Space_is_Subset_of_Tempered_Distribution_Space | https://proofwiki.org/wiki/Lebesgue_1-Space_is_Subset_of_Tempered_Distribution_Space | [
"Tempered Distributions"
] | [
"Definition:Lebesgue Space",
"Definition:Tempered Distribution Space",
"Definition:Tempered Distribution"
] | [
"Definition:Lebesgue Space",
"Definition:P-Seminorm",
"Definition:Schwartz Test Function",
"Definition:Schwartz Space",
"Definition:Functional",
"Integral Operator is Linear",
"Definition:Linear Transformation",
"Definition:Sequence",
"Definition:Schwartz Space",
"Definition:Zero-Limit Sequence in... |
proofwiki-18936 | Characteristic Function of Null Set is A.E. Equal to Zero/Corollary | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $N$ be a $\mu$-null set.
Then:
:$\chi_{X \setminus N} = 1$ $\mu$-almost everywhere.
where $\chi_{X \setminus N}$ is the characteristic function of $X \setminus N$. | From Characteristic Function of Set Difference, we have:
:$\chi_{X \setminus N} = \chi_X - \chi_{X \cap N}$
From Intersection with Subset is Subset, we therefore have:
:$\map {\chi_{X \setminus N} } x = 1 - \map {\chi_N} x$
for each $x \in X$.
From Characteristic Function of Null Set is A.E. Equal to Zero, we have:
... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $N$ be a [[Definition:Null Set|$\mu$-null set]].
Then:
:$\chi_{X \setminus N} = 1$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]].
where $\chi_{X \setminus N}$ is the [[Definition:Characteristic Function of Set|characte... | From [[Characteristic Function of Set Difference]], we have:
:$\chi_{X \setminus N} = \chi_X - \chi_{X \cap N}$
From [[Intersection with Subset is Subset]], we therefore have:
:$\map {\chi_{X \setminus N} } x = 1 - \map {\chi_N} x$
for each $x \in X$.
From [[Characteristic Function of Null Set is A.E. Equal to Z... | Characteristic Function of Null Set is A.E. Equal to Zero/Corollary | https://proofwiki.org/wiki/Characteristic_Function_of_Null_Set_is_A.E._Equal_to_Zero/Corollary | https://proofwiki.org/wiki/Characteristic_Function_of_Null_Set_is_A.E._Equal_to_Zero/Corollary | [
"Characteristic Function of Null Set is A.E. Equal to Zero"
] | [
"Definition:Measure Space",
"Definition:Null Set",
"Definition:Almost Everywhere",
"Definition:Characteristic Function (Set Theory)/Set"
] | [
"Characteristic Function of Set Difference",
"Intersection with Subset is Subset",
"Characteristic Function of Null Set is A.E. Equal to Zero",
"Definition:Almost Everywhere",
"Pointwise Addition preserves A.E. Equality",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Category:Charac... |
proofwiki-18937 | Pointwise Multiplication preserves A.E. Equality | Let $\struct {X, \Sigma, \mu}$ be a measure space.
Let $f, g, h : X \to \overline \R$ be functions with:
:$f = g$ $\mu$-almost everywhere.
Then:
:$f \times h = g \times h$ $\mu$-almost everywhere
where $f \times h$ and $g \times h$ are the pointwise products of $f$ and $h$, and $g$ and $h$ respectively. | Since:
:$f = g$ $\mu$-almost everywhere
there exists a $\mu$-null set $N \subseteq X$ such that:
:if $x \in X$ has $\map f x \ne \map g x$ then $x \in N$.
Note that if $x \in X$ is such that:
:$\map f x = \map g x$
then:
:$\map f x \map h x = \map g x \map h x$
By the Rule of Transposition, we therefore have:
:if $... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Measure Space|measure space]].
Let $f, g, h : X \to \overline \R$ be [[Definition:Extended Real-Valued Function|functions]] with:
:$f = g$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]].
Then:
:$f \times h = g \times h$ [[Definition:Almost Everywhere|$\m... | Since:
:$f = g$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]]
there exists a [[Definition:Null Set|$\mu$-null set]] $N \subseteq X$ such that:
:if $x \in X$ has $\map f x \ne \map g x$ then $x \in N$.
Note that if $x \in X$ is such that:
:$\map f x = \map g x$
then:
:$\map f x \map h x = \map g x \... | Pointwise Multiplication preserves A.E. Equality | https://proofwiki.org/wiki/Pointwise_Multiplication_preserves_A.E._Equality | https://proofwiki.org/wiki/Pointwise_Multiplication_preserves_A.E._Equality | [
"Measure Theory",
"Almost-Everywhere Equality Relation",
"Almost-Everywhere Equality Relation"
] | [
"Definition:Measure Space",
"Definition:Extended Real-Valued Function",
"Definition:Almost Everywhere",
"Definition:Almost Everywhere",
"Definition:Pointwise Multiplication of Extended Real-Valued Functions"
] | [
"Definition:Almost Everywhere",
"Definition:Null Set",
"Rule of Transposition",
"Definition:Null Set",
"Definition:Almost Everywhere",
"Category:Almost-Everywhere Equality Relation"
] |
proofwiki-18938 | Restriction of Measurable Function is Measurable on Trace Sigma-Algebra | Let $\struct {X, \Sigma}$ be a measurable space.
Let $f : X \to \overline \R$ be a $\Sigma$-measurable functions.
Let $E \in \Sigma$.
Let $\Sigma_E$ be the trace $\sigma$-algebra of $E$ in $\Sigma$.
Then the restriction $f \restriction_E$ is $\Sigma_E$-measurable. | From the definition of a $\Sigma_E$-measurable function, we aim to show that:
:$\set {x \in E : \map f x \le \alpha} \in \Sigma_E$
for each $\alpha \in \R$.
Let $\alpha \in \R$.
We have:
:$\set {x \in E : \map f x \le \alpha} = \set {x \in X : \map f x \le \alpha} \cap E$
Since $f$ is $\Sigma$-measurable, we have:
... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $f : X \to \overline \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable functions]].
Let $E \in \Sigma$.
Let $\Sigma_E$ be the [[Definition:Trace Sigma-Algebra|trace $\sigma$-algebra]] of $E$ in $\Sigma$.
Then the [... | From the definition of a [[Definition:Measurable Function|$\Sigma_E$-measurable function]], we aim to show that:
:$\set {x \in E : \map f x \le \alpha} \in \Sigma_E$
for each $\alpha \in \R$.
Let $\alpha \in \R$.
We have:
:$\set {x \in E : \map f x \le \alpha} = \set {x \in X : \map f x \le \alpha} \cap E$
Si... | Restriction of Measurable Function is Measurable on Trace Sigma-Algebra | https://proofwiki.org/wiki/Restriction_of_Measurable_Function_is_Measurable_on_Trace_Sigma-Algebra | https://proofwiki.org/wiki/Restriction_of_Measurable_Function_is_Measurable_on_Trace_Sigma-Algebra | [
"Trace Sigma-Algebras",
"Measurable Functions",
"Trace Sigma-Algebras"
] | [
"Definition:Measurable Space",
"Definition:Measurable Function",
"Definition:Trace Sigma-Algebra",
"Definition:Restriction/Mapping",
"Definition:Measurable Function"
] | [
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Trace Sigma-Algebra",
"Definition:Measurable Function",
"Category:Measurable Functions",
"Category:Trace Sigma-Algebras"
] |
proofwiki-18939 | Variation of Complex Measure is Finite Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.
Let $\cmod \mu$ be the variation of $\mu$.
Then $\cmod \mu$ is a finite measure on $\struct {X, \Sigma}$. | We first show that $\map {\cmod \mu} A \ge 0$ for each $A \in \Sigma$.
Let $A \in \Sigma$.
Let $\map P A$ be the set of finite partitions of $A$ into $\Sigma$-measurable sets.
Then, for each $A \in \Sigma$, we have:
:$\ds \map {\cmod \mu} A = \sup \set {\sum_{j \mathop = 1}^n \cmod {\map \mu {A_j} } : \set {A_1, A_2,... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Complex Measure|complex measure]] on $\struct {X, \Sigma}$.
Let $\cmod \mu$ be the [[Definition:Variation of Complex Measure|variation]] of $\mu$.
Then $\cmod \mu$ is a [[Definition:Finite Measure|finite m... | We first show that $\map {\cmod \mu} A \ge 0$ for each $A \in \Sigma$.
Let $A \in \Sigma$.
Let $\map P A$ be the set of [[Definition:Finite Set|finite]] [[Definition:Set Partition|partitions]] of $A$ into [[Definition:Measurable Set|$\Sigma$-measurable sets]].
Then, for each $A \in \Sigma$, we have:
:$\ds \map {\... | Variation of Complex Measure is Finite Measure | https://proofwiki.org/wiki/Variation_of_Complex_Measure_is_Finite_Measure | https://proofwiki.org/wiki/Variation_of_Complex_Measure_is_Finite_Measure | [
"Variation of Complex Measure is Finite Measure",
"Variation of Complex Measure",
"Finite Measures",
"Complex Measures"
] | [
"Definition:Measurable Space",
"Definition:Complex Measure",
"Definition:Variation/Complex Measure",
"Definition:Finite Measure"
] | [
"Definition:Finite Set",
"Definition:Set Partition",
"Definition:Measurable Set",
"Definition:Supremum of Set",
"Characterization of Measures",
"Definition:Supremum of Set",
"Definition:Measurable Set",
"Definition:Finite",
"Characterization of Measures",
"Definition:Finite",
"Characterization o... |
proofwiki-18940 | Measures in Jordan Decomposition of Complex Measure are Finite | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.
Let $\tuple {\mu_1, \mu_2, \mu_3, \mu_4}$ be the Jordan decomposition of $\mu$.
Then:
:$\mu_1$, $\mu_2$, $\mu_3$ and $\mu_4$ are finite. | Let $\mu_R$ be the real part of $\mu$.
Let $\mu_I$ be the imaginary part of $\mu$.
Then:
:$\tuple {\mu_1, \mu_2}$ is the Jordan decomposition of $\mu_R$
and:
:$\tuple {\mu_3, \mu_4}$ is the Jordan decomposition of $\mu_I$.
From the definition of the real part and imaginary part, we have that both $\mu_R$ and $\mu_I$ ar... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Complex Measure|complex measure]] on $\struct {X, \Sigma}$.
Let $\tuple {\mu_1, \mu_2, \mu_3, \mu_4}$ be the [[Definition:Jordan Decomposition of Complex Measure|Jordan decomposition]] of $\mu$.
Then:
:$\m... | Let $\mu_R$ be the [[Definition:Real Part of Complex Measure|real part]] of $\mu$.
Let $\mu_I$ be the [[Definition:Imaginary Part of Complex Measure|imaginary part]] of $\mu$.
Then:
:$\tuple {\mu_1, \mu_2}$ is the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu_R$
and:
:$\tuple {\mu_3, \mu_4}$ is ... | Measures in Jordan Decomposition of Complex Measure are Finite | https://proofwiki.org/wiki/Measures_in_Jordan_Decomposition_of_Complex_Measure_are_Finite | https://proofwiki.org/wiki/Measures_in_Jordan_Decomposition_of_Complex_Measure_are_Finite | [
"Complex Measures"
] | [
"Definition:Measurable Space",
"Definition:Complex Measure",
"Definition:Jordan Decomposition of Complex Measure",
"Definition:Finite Measure"
] | [
"Definition:Real Part of Complex Measure",
"Definition:Imaginary Part of Complex Measure",
"Definition:Jordan Decomposition",
"Definition:Jordan Decomposition",
"Definition:Real Part of Complex Measure",
"Definition:Imaginary Part of Complex Measure",
"Definition:Finite Measure/Signed Measure",
"Jorda... |
proofwiki-18941 | Absolute Continuity of Signed Measure in terms of Jordan Decomposition | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a measure on $\struct {X, \Sigma}$.
Let $\nu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\tuple {\nu^+, \nu^-}$ be the Jordan decomposition of $\nu$.
Then $\nu$ is absolutely continuous with respect to $\mu$ {{iff}}:
:$\nu^+$ and $\nu^-$ are absolute... | We have that $\nu$ is absolutely continuous with respect to $\mu$ {{iff}}:
:$\size \nu$ is absolutely continuous with respect to $\mu$
where $\size \nu$ is the variation of $\nu$.
From the definition of variation, we have:
:$\size \nu = \nu^+ + \nu^-$
Suppose that $\nu$ is absolutely continuous with respect to $\mu$.... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Measure (Measure Theory)|measure]] on $\struct {X, \Sigma}$.
Let $\nu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $\tuple {\nu^+, \nu^-}$ be the [[Definition:Jordan Decomp... | We have that $\nu$ is [[Definition:Absolute Continuity/Signed Measure|absolutely continuous]] with respect to $\mu$ {{iff}}:
:$\size \nu$ is [[Definition:Absolutely Continuous Measure|absolutely continuous]] with respect to $\mu$
where $\size \nu$ is the [[Definition:Variation of Signed Measure|variation]] of $\nu$.
... | Absolute Continuity of Signed Measure in terms of Jordan Decomposition | https://proofwiki.org/wiki/Absolute_Continuity_of_Signed_Measure_in_terms_of_Jordan_Decomposition | https://proofwiki.org/wiki/Absolute_Continuity_of_Signed_Measure_in_terms_of_Jordan_Decomposition | [
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Measure (Measure Theory)",
"Definition:Signed Measure",
"Definition:Jordan Decomposition",
"Definition:Absolute Continuity/Signed Measure",
"Definition:Absolute Continuity/Measure"
] | [
"Definition:Absolute Continuity/Signed Measure",
"Definition:Absolute Continuity/Measure",
"Definition:Variation/Signed Measure",
"Definition:Variation/Signed Measure",
"Definition:Absolute Continuity/Signed Measure",
"Definition:Absolute Continuity/Measure",
"Definition:Absolute Continuity/Measure",
... |
proofwiki-18942 | Variation of Signed Measure is Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\size \mu$ be the variation of $\mu$.
Then $\size \mu$ is a measure. | Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.
Then:
:$\size \mu = \mu^+ + \mu^-$
So $\size \mu$ is a measure from Linear Combination of Measures.
{{qed}}
Category:Measures
Category:Variation of Signed Measure
4i2hsl589zdql24c6q7ih9emlqzjepj | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $\size \mu$ be the [[Definition:Variation of Signed Measure|variation]] of $\mu$.
Then $\size \mu$ is a [[Definition:Measure (Measure Theory)|me... | Let $\tuple {\mu^+, \mu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$.
Then:
:$\size \mu = \mu^+ + \mu^-$
So $\size \mu$ is a [[Definition:Measure (Measure Theory)|measure]] from [[Linear Combination of Measures]].
{{qed}}
[[Category:Measures]]
[[Category:Variation of Signed Measure... | Variation of Signed Measure is Measure | https://proofwiki.org/wiki/Variation_of_Signed_Measure_is_Measure | https://proofwiki.org/wiki/Variation_of_Signed_Measure_is_Measure | [
"Variation of Signed Measure",
"Signed Measures",
"Measures",
"Measures",
"Variation of Signed Measure"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Variation/Signed Measure",
"Definition:Measure (Measure Theory)"
] | [
"Definition:Jordan Decomposition",
"Definition:Measure (Measure Theory)",
"Linear Combination of Measures",
"Category:Measures",
"Category:Variation of Signed Measure"
] |
proofwiki-18943 | Characterization of Null Sets of Variation of Signed Measure | Let $\struct {X, \Sigma}$ be measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Let $\size \mu$ be the variation of $\mu$.
Then $A \in \Sigma$ is such that $\map {\size \mu} A = 0$ {{iff}}:
:for each $\Sigma$-measurable set $B \subseteq A$, we have $\map \mu B = 0$. | Let $\tuple {\mu^+, \mu^-}$ be the Jordan decomposition of $\mu$.
Then, from the definition of variation:
:$\size \mu = \mu^+ + \mu^-$
Suppose that $A \in \Sigma$ is such that $\map {\size \mu} A = 0$.
Then:
:$\map {\mu^+} A + \map {\mu^-} A = 0$
Since $\mu^+ \ge 0$ and $\mu^- \ge 0$ we have:
:$\map {\mu^+} A = 0 $ ... | Let $\struct {X, \Sigma}$ be [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Let $\size \mu$ be the [[Definition:Variation of Signed Measure|variation]] of $\mu$.
Then $A \in \Sigma$ is such that $\map {\size \mu} A = 0$ {{iff... | Let $\tuple {\mu^+, \mu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\mu$.
Then, from the definition of [[Definition:Variation of Signed Measure|variation]]:
:$\size \mu = \mu^+ + \mu^-$
Suppose that $A \in \Sigma$ is such that $\map {\size \mu} A = 0$.
Then:
:$\map {\mu^+} A + \map {... | Characterization of Null Sets of Variation of Signed Measure | https://proofwiki.org/wiki/Characterization_of_Null_Sets_of_Variation_of_Signed_Measure | https://proofwiki.org/wiki/Characterization_of_Null_Sets_of_Variation_of_Signed_Measure | [
"Variation of Signed Measure"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Variation/Signed Measure",
"Definition:Measurable Set"
] | [
"Definition:Jordan Decomposition",
"Definition:Variation/Signed Measure",
"Null Sets Closed under Subset",
"Definition:Measurable Set",
"Definition:Measurable Set",
"Definition:Measurable Set",
"Definition:Jordan Decomposition",
"Definition:Measurable Set",
"Definition:Measurable Set",
"Definition... |
proofwiki-18944 | Signed Measure may not be Monotone | Let $\struct {X, \Sigma}$ be measurable space.
Let $\mu$ be a signed measure on $\struct {X, \Sigma}$.
Then $\mu$ may not be monotone. | Let:
:$\struct {X, \Sigma} = \struct {\R, \map \BB \R}$
where $\map \BB \R$ is the Borel $\sigma$-algebra on $\R$.
Define:
:$\mu = \delta_1 - 2 \delta_2$
where $\delta_1$ and $\delta_2$ are the Dirac measures at $1$ and $2$ respectively.
Since $\delta_1$ and $\delta_2$ are both finite measures, we have:
:$\mu$ is a s... | Let $\struct {X, \Sigma}$ be [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Then $\mu$ may not be [[Definition:Monotone (Measure Theory)|monotone]]. | Let:
:$\struct {X, \Sigma} = \struct {\R, \map \BB \R}$
where $\map \BB \R$ is the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] on $\R$.
Define:
:$\mu = \delta_1 - 2 \delta_2$
where $\delta_1$ and $\delta_2$ are the [[Definition:Dirac Measure|Dirac measures]] at $1$ and $2$ respectively.
Since $\del... | Signed Measure may not be Monotone | https://proofwiki.org/wiki/Signed_Measure_may_not_be_Monotone | https://proofwiki.org/wiki/Signed_Measure_may_not_be_Monotone | [
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Signed Measure",
"Definition:Monotone (Measure Theory)"
] | [
"Definition:Borel Sigma-Algebra",
"Definition:Dirac Measure",
"Definition:Finite Measure",
"Definition:Signed Measure",
"Linear Combination of Signed Measures is Signed Measure",
"Definition:Monotone (Measure Theory)",
"Category:Signed Measures"
] |
proofwiki-18945 | Sigma-Algebra Closed under Set Difference | Let $\struct {X, \Sigma}$ be a measurable space.
Let $A, B \in \Sigma$.
Then the set difference $A \setminus B$ is contained in $\Sigma$. | Since $\sigma$-algebras are closed under relative complement, we have:
:$\relcomp X B \in \Sigma$
By Sigma-Algebra Closed under Finite Intersection, we have:
:$A \cap \relcomp X B \in \Sigma$
From Set Difference as Intersection with Relative Complement, we have:
:$A \setminus B = A \cap \relcomp X B$
so:
:$A \setmin... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $A, B \in \Sigma$.
Then the [[Definition:Set Difference|set difference]] $A \setminus B$ is contained in $\Sigma$. | Since [[Definition:Sigma-Algebra|$\sigma$-algebras]] are [[Definition:Closed under Mapping|closed]] under [[Definition:Relative Complement|relative complement]], we have:
:$\relcomp X B \in \Sigma$
By [[Sigma-Algebra Closed under Finite Intersection]], we have:
:$A \cap \relcomp X B \in \Sigma$
From [[Set Differe... | Sigma-Algebra Closed under Set Difference | https://proofwiki.org/wiki/Sigma-Algebra_Closed_under_Set_Difference | https://proofwiki.org/wiki/Sigma-Algebra_Closed_under_Set_Difference | [
"Sigma-Algebras",
"Set Difference"
] | [
"Definition:Measurable Space",
"Definition:Set Difference"
] | [
"Definition:Sigma-Algebra",
"Definition:Closed under Mapping",
"Definition:Relative Complement",
"Sigma-Algebra Closed under Finite Intersection",
"Set Difference as Intersection with Relative Complement",
"Category:Sigma-Algebras",
"Category:Set Difference"
] |
proofwiki-18946 | Sigma-Algebra Closed under Symmetric Difference | Let $\struct {X, \Sigma}$ be a measurable space.
Let $A, B \in \Sigma$.
Then the symmetric difference $A \Delta B$ is contained in $\Sigma$. | From Sigma-Algebra Closed under Set Difference, we have:
:$A \setminus B \in \Sigma$
and:
:$B \setminus A \in \Sigma$
Since $\sigma$-algebras are closed under countable union, we have:
:$\paren {A \setminus B} \cup \paren {B \setminus A} \in \Sigma$
From the definition of symmetric difference, we have:
:$A \Delta B ... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $A, B \in \Sigma$.
Then the [[Definition:Symmetric Difference|symmetric difference]] $A \Delta B$ is contained in $\Sigma$. | From [[Sigma-Algebra Closed under Set Difference]], we have:
:$A \setminus B \in \Sigma$
and:
:$B \setminus A \in \Sigma$
Since [[Definition:Sigma-Algebra|$\sigma$-algebras]] are [[Definition:Closed under Mapping|closed]] under [[Definition:Countable Union|countable union]], we have:
:$\paren {A \setminus B} \cu... | Sigma-Algebra Closed under Symmetric Difference | https://proofwiki.org/wiki/Sigma-Algebra_Closed_under_Symmetric_Difference | https://proofwiki.org/wiki/Sigma-Algebra_Closed_under_Symmetric_Difference | [
"Sigma-Algebras",
"Symmetric Difference"
] | [
"Definition:Measurable Space",
"Definition:Symmetric Difference"
] | [
"Sigma-Algebra Closed under Set Difference",
"Definition:Sigma-Algebra",
"Definition:Closed under Mapping",
"Definition:Set Union/Countable Union",
"Definition:Symmetric Difference",
"Category:Sigma-Algebras",
"Category:Symmetric Difference"
] |
proofwiki-18947 | Characterization of Null Sets of Variation of Complex Measure | Let $\struct {X, \Sigma}$ be measurable space.
Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.
Let $\size \mu$ be the variation of $\mu$.
Then $A \in \Sigma$ is such that $\map {\size \mu} A = 0$ {{iff}}:
:for each $\Sigma$-measurable set $B \subseteq A$, we have $\map \mu B = 0$. | Let $A \in \Sigma$.
Let $\map P A$ be the set of finite partitions of $A$ into $\Sigma$-measurable sets.
From the definition of variation, we have:
:$\ds \map {\cmod \mu} A = \sup \set {\sum_{j \mathop = 1}^n \cmod {\map \mu {A_j} } : \set {A_1, A_2, \ldots, A_n} \in \map P A}$
Suppose that:
:for each $\Sigma$-measura... | Let $\struct {X, \Sigma}$ be [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Complex Measure|complex measure]] on $\struct {X, \Sigma}$.
Let $\size \mu$ be the [[Definition:Variation of Complex Measure|variation]] of $\mu$.
Then $A \in \Sigma$ is such that $\map {\size \mu} A = 0$ {{... | Let $A \in \Sigma$.
Let $\map P A$ be the set of [[Definition:Finite Set|finite]] [[Definition:Set Partition|partitions]] of $A$ into [[Definition:Measurable Set|$\Sigma$-measurable sets]].
From the definition of [[Definition:Variation of Complex Measure|variation]], we have:
:$\ds \map {\cmod \mu} A = \sup \set {\s... | Characterization of Null Sets of Variation of Complex Measure | https://proofwiki.org/wiki/Characterization_of_Null_Sets_of_Variation_of_Complex_Measure | https://proofwiki.org/wiki/Characterization_of_Null_Sets_of_Variation_of_Complex_Measure | [
"Complex Measures",
"Variation of Complex Measure",
"Variation of Complex Measure"
] | [
"Definition:Measurable Space",
"Definition:Complex Measure",
"Definition:Variation/Complex Measure",
"Definition:Measurable Set"
] | [
"Definition:Finite Set",
"Definition:Set Partition",
"Definition:Measurable Set",
"Definition:Variation/Complex Measure",
"Definition:Measurable Set",
"Definition:Supremum of Set/Real Numbers",
"Definition:Positive/Real Number",
"Definition:Measurable Set",
"Sigma-Algebra Closed under Set Difference... |
proofwiki-18948 | Absolute Continuity of Complex Measure in terms of Jordan Decomposition | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a measure on $\struct {X, \Sigma}$.
Let $\nu$ be a complex measure on $\struct {X, \Sigma}$.
Let $\tuple {\nu_1, \nu_2, \nu_3, \nu_4}$ be the Jordan decomposition of $\nu$.
Then $\nu$ is absolutely continuous with respect to $\mu$ {{iff}}:
:$\nu_1$, $\nu_2$,... | Let $\cmod \nu$ be the variation of $\nu$. | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Measure (Measure Theory)|measure]] on $\struct {X, \Sigma}$.
Let $\nu$ be a [[Definition:Complex Measure|complex measure]] on $\struct {X, \Sigma}$.
Let $\tuple {\nu_1, \nu_2, \nu_3, \nu_4}$ be the [[Definiti... | Let $\cmod \nu$ be the [[Definition:Variation of Complex Measure|variation]] of $\nu$. | Absolute Continuity of Complex Measure in terms of Jordan Decomposition | https://proofwiki.org/wiki/Absolute_Continuity_of_Complex_Measure_in_terms_of_Jordan_Decomposition | https://proofwiki.org/wiki/Absolute_Continuity_of_Complex_Measure_in_terms_of_Jordan_Decomposition | [
"Complex Measures",
"Absolutely Continuous Measures"
] | [
"Definition:Measurable Space",
"Definition:Measure (Measure Theory)",
"Definition:Complex Measure",
"Definition:Jordan Decomposition of Complex Measure",
"Definition:Absolute Continuity/Complex Measure",
"Definition:Absolute Continuity/Measure"
] | [
"Definition:Variation/Complex Measure"
] |
proofwiki-18949 | Bound for Variation of Complex Measure in terms of Jordan Decomposition | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a complex measure on $\struct {X, \Sigma}$.
Let $\cmod \mu$ be the variation of $\mu$.
Let $\tuple {\mu_1, \mu_2, \mu_3, \mu_4}$ be the Jordan decomposition of $\mu$.
Then:
:$\map {\cmod \mu} A \le \map {\mu_1} A + \map {\mu_2} A + \map {\mu_3} A + \map {... | Let $A \in \Sigma$.
Let $\map P A$ be the set of finite partitions of $A$ into $\Sigma$-measurable sets.
From the definition of variation, we have:
:$\ds \map {\cmod \mu} A = \sup \set {\sum_{j \mathop = 1}^n \cmod {\map \mu {A_j} } : \set {A_1, A_2, \ldots, A_n} \in \map P A}$
Let:
:$\set {A_1, A_2, \ldots, A_n} \in... | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Complex Measure|complex measure]] on $\struct {X, \Sigma}$.
Let $\cmod \mu$ be the [[Definition:Variation of Complex Measure|variation]] of $\mu$.
Let $\tuple {\mu_1, \mu_2, \mu_3, \mu_4}$ be the [[Definiti... | Let $A \in \Sigma$.
Let $\map P A$ be the set of [[Definition:Finite Set|finite]] [[Definition:Set Partition|partitions]] of $A$ into [[Definition:Measurable Set|$\Sigma$-measurable sets]].
From the definition of [[Definition:Variation of Complex Measure|variation]], we have:
:$\ds \map {\cmod \mu} A = \sup \set {\... | Bound for Variation of Complex Measure in terms of Jordan Decomposition | https://proofwiki.org/wiki/Bound_for_Variation_of_Complex_Measure_in_terms_of_Jordan_Decomposition | https://proofwiki.org/wiki/Bound_for_Variation_of_Complex_Measure_in_terms_of_Jordan_Decomposition | [
"Complex Measures"
] | [
"Definition:Measurable Space",
"Definition:Complex Measure",
"Definition:Variation/Complex Measure",
"Definition:Jordan Decomposition of Complex Measure"
] | [
"Definition:Finite Set",
"Definition:Set Partition",
"Definition:Measurable Set",
"Definition:Variation/Complex Measure",
"Definition:Jordan Decomposition of Complex Measure",
"Triangle Inequality/Complex Numbers",
"Definition:Measure (Measure Theory)",
"Definition:Set Partition",
"Definition:Pairwi... |
proofwiki-18950 | Characterization of Absolute Continuity of Signed Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a measure on $\struct {X, \Sigma}$.
Let $\nu$ be a signed measure on $\struct {X, \Sigma}$.
Then $\nu$ is absolutely continuous with respect to $\mu$ {{iff}}:
:for all $A \in \Sigma$ with $\map \mu A = 0$, we have $\map \nu A = 0$. | Let $\tuple {\nu^+, \nu^-}$ be the Jordan decomposition of $\nu$.
Let $\size \nu$ be the variation of $\nu$. | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Measure (Measure Theory)|measure]] on $\struct {X, \Sigma}$.
Let $\nu$ be a [[Definition:Signed Measure|signed measure]] on $\struct {X, \Sigma}$.
Then $\nu$ is [[Definition:Absolutely Continuous Signed Meas... | Let $\tuple {\nu^+, \nu^-}$ be the [[Definition:Jordan Decomposition|Jordan decomposition]] of $\nu$.
Let $\size \nu$ be the [[Definition:Variation of Signed Measure|variation]] of $\nu$. | Characterization of Absolute Continuity of Signed Measure | https://proofwiki.org/wiki/Characterization_of_Absolute_Continuity_of_Signed_Measure | https://proofwiki.org/wiki/Characterization_of_Absolute_Continuity_of_Signed_Measure | [
"Absolutely Continuous Signed Measures",
"Absolutely Continuous Measures",
"Signed Measures"
] | [
"Definition:Measurable Space",
"Definition:Measure (Measure Theory)",
"Definition:Signed Measure",
"Definition:Absolute Continuity/Signed Measure"
] | [
"Definition:Jordan Decomposition",
"Definition:Variation/Signed Measure",
"Definition:Jordan Decomposition"
] |
proofwiki-18951 | Characterization of Absolute Continuity of Complex Measure | Let $\struct {X, \Sigma}$ be a measurable space.
Let $\mu$ be a measure on $\struct {X, \Sigma}$.
Let $\nu$ be a complex measure on $\struct {X, \Sigma}$.
Then $\nu$ is absolutely continuous with respect to $\mu$ {{iff}}:
:for all $A \in \Sigma$ with $\map \mu A = 0$, we have $\map \nu A = 0$. | Let $\tuple {\nu_1, \nu_2, \nu_3, \nu_4}$ be the Jordan decomposition of $\nu$.
Let $\size \nu$ be the variation of $\nu$. | Let $\struct {X, \Sigma}$ be a [[Definition:Measurable Space|measurable space]].
Let $\mu$ be a [[Definition:Measure (Measure Theory)|measure]] on $\struct {X, \Sigma}$.
Let $\nu$ be a [[Definition:Complex Measure|complex measure]] on $\struct {X, \Sigma}$.
Then $\nu$ is [[Definition:Absolute Continuity/Complex Mea... | Let $\tuple {\nu_1, \nu_2, \nu_3, \nu_4}$ be the [[Definition:Jordan Decomposition of Complex Measure|Jordan decomposition]] of $\nu$.
Let $\size \nu$ be the [[Definition:Variation of Complex Measure|variation]] of $\nu$. | Characterization of Absolute Continuity of Complex Measure | https://proofwiki.org/wiki/Characterization_of_Absolute_Continuity_of_Complex_Measure | https://proofwiki.org/wiki/Characterization_of_Absolute_Continuity_of_Complex_Measure | [
"Complex Measures"
] | [
"Definition:Measurable Space",
"Definition:Measure (Measure Theory)",
"Definition:Complex Measure",
"Definition:Absolute Continuity/Complex Measure"
] | [
"Definition:Jordan Decomposition of Complex Measure",
"Definition:Variation/Complex Measure",
"Definition:Jordan Decomposition of Complex Measure"
] |
proofwiki-18952 | Characteristic Function of Disjoint Union/Corollary | Let $\set {D_1, D_2, \ldots, D_N}$ be a set of pairwise disjoint subsets of $X$.
Let:
:$\ds D = \bigcup_{n \mathop = 1}^N D_n$
Then:
:$\ds \chi_D = \sum_{n \mathop = 1}^N \chi_{D_n}$
where:
:$\chi_D$ is the characteristic function of $D$
:$\chi_{D_n}$ is the characteristic function of $D_n$. | We can extend $\set {D_1, D_2, \ldots, D_N}$ to a sequence $\sequence {D_n}_{n \mathop \in \N}$ of subsets of $X$ by setting:
:$D_i = \O$ for $i \ge N + 1$
Clearly, from Intersection with Empty Set, we have:
:$D_i \cap D_j = \O$ for $i \ge N + 1$ and all $j$.
So $\sequence {D_n}_{n \mathop \in \N}$ is a sequence of ... | Let $\set {D_1, D_2, \ldots, D_N}$ be a [[Definition:Set|set]] of [[Definition:Pairwise Disjoint|pairwise disjoint]] subsets of $X$.
Let:
:$\ds D = \bigcup_{n \mathop = 1}^N D_n$
Then:
:$\ds \chi_D = \sum_{n \mathop = 1}^N \chi_{D_n}$
where:
:$\chi_D$ is the [[Definition:Characteristic Function (Set Theory)|c... | We can extend $\set {D_1, D_2, \ldots, D_N}$ to a [[Definition:Sequence|sequence]] $\sequence {D_n}_{n \mathop \in \N}$ of subsets of $X$ by setting:
:$D_i = \O$ for $i \ge N + 1$
Clearly, from [[Intersection with Empty Set]], we have:
:$D_i \cap D_j = \O$ for $i \ge N + 1$ and all $j$.
So $\sequence {D_n}_{n \m... | Characteristic Function of Disjoint Union/Corollary | https://proofwiki.org/wiki/Characteristic_Function_of_Disjoint_Union/Corollary | https://proofwiki.org/wiki/Characteristic_Function_of_Disjoint_Union/Corollary | [
"Characteristic Function of Disjoint Union"
] | [
"Definition:Set",
"Definition:Pairwise Disjoint",
"Definition:Characteristic Function (Set Theory)",
"Definition:Characteristic Function (Set Theory)"
] | [
"Definition:Sequence",
"Intersection with Empty Set",
"Definition:Sequence",
"Definition:Pairwise Disjoint",
"Characteristic Function of Disjoint Union",
"Characteristic Function of Empty Set",
"Category:Characteristic Function of Disjoint Union"
] |
proofwiki-18953 | Integral of Positive Function with respect to Counting Measure on Natural Numbers | Consider the measure space $\struct {\N, \powerset \N, \mu}$ where $\mu$ is the counting measure on $\struct {\N, \powerset \N}$.
Let $f : \N \to \R$ be a function.
Then:
:$\ds \int f \rd \mu = \sum_{n \mathop = 1}^\infty \map f n$ | Clearly we have:
:$\set {x \in \N : \map f x \le \alpha} \in \powerset \N$
for each $\alpha \in \R$, so any function $f : \N \to \R$ is $\powerset \N$-measurable.
Similarly, an arbitrary subset of $\N$ is clearly $\powerset \N$-measurable.
For each $n \in \N$, define $f_n : \N \to \R$ by:
:$\ds \map {f_n} k = \begin{... | Consider the [[Definition:Measure Space|measure space]] $\struct {\N, \powerset \N, \mu}$ where $\mu$ is the [[Definition:Counting Measure|counting measure]] on $\struct {\N, \powerset \N}$.
Let $f : \N \to \R$ be a [[Definition:Function|function]].
Then:
:$\ds \int f \rd \mu = \sum_{n \mathop = 1}^\infty \map f ... | Clearly we have:
:$\set {x \in \N : \map f x \le \alpha} \in \powerset \N$
for each $\alpha \in \R$, so any [[Definition:Function|function]] $f : \N \to \R$ is [[Definition:Measurable Function|$\powerset \N$-measurable]].
Similarly, an arbitrary subset of $\N$ is clearly [[Definition:Measurable Function|$\powerset ... | Integral of Positive Function with respect to Counting Measure on Natural Numbers | https://proofwiki.org/wiki/Integral_of_Positive_Function_with_respect_to_Counting_Measure_on_Natural_Numbers | https://proofwiki.org/wiki/Integral_of_Positive_Function_with_respect_to_Counting_Measure_on_Natural_Numbers | [
"Counting Measure"
] | [
"Definition:Measure Space",
"Definition:Counting Measure",
"Definition:Function"
] | [
"Definition:Function",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Definition:Increasing Sequence of Real-Valued Functions",
"Definition:Increasing Sequence of Extended Real-Valued Functions",
"Tail of Convergent Sequence",
"Definition:Increasing Sequence of Extended Real-Valued... |
proofwiki-18954 | In Connected Smooth Manifold Any Two Points can be Joined by Admissible Curve | Let $M$ be a connected smooth manifold with or without a boundary.
Let $p, q \in M$ be points.
Let $\gamma : \closedint a b \to M$ be an admissible curve.
Then:
:$\forall p, q \in M : \exists \gamma \subset M : \paren {\map \gamma a = p} \land \paren {\map \gamma b = q}$ | For $p, q \in M$, we write:
:$p \sim q$
{{iff}} there exists an admissible curve $\gamma : \closedint a b \to M$ such that $\map \gamma a = p$ and $\map \gamma b = q$ | Let $M$ be a connected smooth manifold with or without a boundary.
Let $p, q \in M$ be [[Definition:Point|points]].
Let $\gamma : \closedint a b \to M$ be an [[Definition:Admissible Curve|admissible curve]].
Then:
:$\forall p, q \in M : \exists \gamma \subset M : \paren {\map \gamma a = p} \land \paren {\map \gamm... | For $p, q \in M$, we write:
:$p \sim q$
{{iff}} there exists an [[Definition:Admissible Curve|admissible curve]] $\gamma : \closedint a b \to M$ such that $\map \gamma a = p$ and $\map \gamma b = q$ | In Connected Smooth Manifold Any Two Points can be Joined by Admissible Curve | https://proofwiki.org/wiki/In_Connected_Smooth_Manifold_Any_Two_Points_can_be_Joined_by_Admissible_Curve | https://proofwiki.org/wiki/In_Connected_Smooth_Manifold_Any_Two_Points_can_be_Joined_by_Admissible_Curve | [
"Riemannian Geometry"
] | [
"Definition:Point",
"Definition:Piecewise Regular Curve Segment"
] | [
"Definition:Piecewise Regular Curve Segment",
"Definition:Piecewise Regular Curve Segment"
] |
proofwiki-18955 | Function Measurable with respect to Power Set | Let $\struct {X, \powerset X}$ be a measurable space, where $\powerset X$ is the power set of $X$.
Let $f : X \to \overline \R$ be a function.
Then $f$ is $\powerset X$-measurable function. | For each $\alpha \in \R$, we have:
:$\set {x \in X : \map f x \le \alpha} \subseteq X$
That is, from the definition of power set:
:$\set {x \in X : \map f x \le \alpha} \in \powerset X$
So for each $\alpha \in \R$:
:the set $\set {x \in X : \map f x \le \alpha}$ is $\powerset X$-measurable.
So:
:$f$ is $\powerset X$-m... | Let $\struct {X, \powerset X}$ be a [[Definition:Measurable Space|measurable space]], where $\powerset X$ is the [[Definition:Power Set|power set]] of $X$.
Let $f : X \to \overline \R$ be a [[Definition:Extended Real-Valued Function|function]].
Then $f$ is [[Definition:Measurable Function|$\powerset X$-measurable]]... | For each $\alpha \in \R$, we have:
:$\set {x \in X : \map f x \le \alpha} \subseteq X$
That is, from the definition of [[Definition:Power Set|power set]]:
:$\set {x \in X : \map f x \le \alpha} \in \powerset X$
So for each $\alpha \in \R$:
:the set $\set {x \in X : \map f x \le \alpha}$ is [[Definition:Measurable... | Function Measurable with respect to Power Set | https://proofwiki.org/wiki/Function_Measurable_with_respect_to_Power_Set | https://proofwiki.org/wiki/Function_Measurable_with_respect_to_Power_Set | [
"Measurable Functions"
] | [
"Definition:Measurable Space",
"Definition:Power Set",
"Definition:Extended Real-Valued Function",
"Definition:Measurable Function"
] | [
"Definition:Power Set",
"Definition:Measurable Set",
"Definition:Measurable Set",
"Category:Measurable Functions"
] |
proofwiki-18956 | Counting Measure on Natural Numbers is Sigma-Finite | Let $\mu$ be the counting measure on $\struct {\N, \powerset \N}$.
Then $\mu$ is $\sigma$-finite. | For each $n \in \N$, define:
:$X_n = \N \cap \closedint 1 n = \set {1, 2, \ldots, n}$
Since $X_n \subseteq \N$ for each $n$ from Intersection is Subset, we have that:
:$X_n$ is $\powerset \N$-measurable for each $n$.
We show that $\sequence {X_n}_{n \mathop \in \N}$ is an exhausting sequence in $\powerset \N$ and tha... | Let $\mu$ be the [[Definition:Counting Measure|counting measure]] on $\struct {\N, \powerset \N}$.
Then $\mu$ is [[Definition:Sigma-Finite Measure|$\sigma$-finite]]. | For each $n \in \N$, define:
:$X_n = \N \cap \closedint 1 n = \set {1, 2, \ldots, n}$
Since $X_n \subseteq \N$ for each $n$ from [[Intersection is Subset]], we have that:
:$X_n$ is [[Definition:Measurable Set|$\powerset \N$-measurable]] for each $n$.
We show that $\sequence {X_n}_{n \mathop \in \N}$ is an [[Defi... | Counting Measure on Natural Numbers is Sigma-Finite | https://proofwiki.org/wiki/Counting_Measure_on_Natural_Numbers_is_Sigma-Finite | https://proofwiki.org/wiki/Counting_Measure_on_Natural_Numbers_is_Sigma-Finite | [
"Counting Measure"
] | [
"Definition:Counting Measure",
"Definition:Sigma-Finite Measure"
] | [
"Intersection is Subset",
"Definition:Measurable Set",
"Definition:Exhausting Sequence of Sets",
"Definition:Counting Measure",
"Set Intersection Preserves Subsets",
"Definition:Increasing Sequence of Sets",
"Definition:Measurable Set",
"Definition:Exhausting Sequence of Sets",
"Definition:Sigma-Fin... |
proofwiki-18957 | Function A.E. Equal to Measurable Function in Complete Measure Space is Measurable | Let $\struct {X, \Sigma, \mu}$ be a complete measure space.
Let $f : X \to \overline \R$ be a $\Sigma$-measurable function.
Let $g : X \to \overline \R$ be a function such that:
:$f = g$ $\mu$-almost everywhere.
Then $g$ is $\Sigma$-measurable. | We aim to show that:
:$\set {x \in X : \map g x \le \alpha} \in \Sigma$
for each $\alpha \in \R$.
Let $\alpha \in \R$.
Since $f = g$ $\mu$-almost everywhere there exists a $\mu$-null set such that:
:whenever $x \in X$ has $\map f x \ne \map g x$, we have $x \in N$.
We have:
{{begin-eqn}}
{{eqn | l = \set {x \in X ... | Let $\struct {X, \Sigma, \mu}$ be a [[Definition:Complete Measure Space|complete measure space]].
Let $f : X \to \overline \R$ be a [[Definition:Measurable Function|$\Sigma$-measurable function]].
Let $g : X \to \overline \R$ be a [[Definition:Extended Real-Valued Function|function]] such that:
:$f = g$ [[Definitio... | We aim to show that:
:$\set {x \in X : \map g x \le \alpha} \in \Sigma$
for each $\alpha \in \R$.
Let $\alpha \in \R$.
Since $f = g$ [[Definition:Almost Everywhere|$\mu$-almost everywhere]] there exists a [[Definition:Null Set|$\mu$-null set]] such that:
:whenever $x \in X$ has $\map f x \ne \map g x$, we have... | Function A.E. Equal to Measurable Function in Complete Measure Space is Measurable | https://proofwiki.org/wiki/Function_A.E._Equal_to_Measurable_Function_in_Complete_Measure_Space_is_Measurable | https://proofwiki.org/wiki/Function_A.E._Equal_to_Measurable_Function_in_Complete_Measure_Space_is_Measurable | [
"Complete Measure Spaces"
] | [
"Definition:Complete Measure Space",
"Definition:Measurable Function",
"Definition:Extended Real-Valued Function",
"Definition:Almost Everywhere",
"Definition:Measurable Function"
] | [
"Definition:Almost Everywhere",
"Definition:Null Set",
"Intersection with Subset is Subset",
"Union with Complement",
"Union Distributes over Intersection",
"Definition:Measurable Function",
"Definition:Sigma-Algebra",
"Definition:Closed under Mapping",
"Definition:Relative Complement",
"Sigma-Alg... |
proofwiki-18958 | Lower Sum of Refinement | Let $\closedint a b$ be a closed interval.
Let $P$ be a finite subdivision of $\closedint a b$.
Let $Q$ be a refinement of $P$.
Then:
:$\map L {f, P} \le \map L {f, Q}$
where $\map L {f, P}$ and $\map L {f, Q}$ denotes the lower Darboux sum of $f$ with respect to $P$ and $Q$. | Write:
:$P = \set {x_0, x_1, \ldots, x_k}$
and:
:$Q = \set {y_0, y_1, \ldots, y_l}$
where:
:$a = x_0 < x_1 < \ldots < x_k = b$
and:
:$a = y_0 < y_1 < \ldots < y_l = b$
Since $P \subseteq Q$, we have $k \le l$ from Cardinality of Subset of Finite Set.
Set:
:$m_i = \inf \set {\map f x : x \in \closedint {x_{i - 1} } {x... | Let $\closedint a b$ be a [[Definition:Closed Interval|closed interval]].
Let $P$ be a [[Definition:Finite Subdivision|finite subdivision]] of $\closedint a b$.
Let $Q$ be a [[Definition:Refinement of Finite Subdivision|refinement]] of $P$.
Then:
:$\map L {f, P} \le \map L {f, Q}$
where $\map L {f, P}$ and $\m... | Write:
:$P = \set {x_0, x_1, \ldots, x_k}$
and:
:$Q = \set {y_0, y_1, \ldots, y_l}$
where:
:$a = x_0 < x_1 < \ldots < x_k = b$
and:
:$a = y_0 < y_1 < \ldots < y_l = b$
Since $P \subseteq Q$, we have $k \le l$ from [[Cardinality of Subset of Finite Set]].
Set:
:$m_i = \inf \set {\map f x : x \in \closedint {... | Lower Sum of Refinement | https://proofwiki.org/wiki/Lower_Sum_of_Refinement | https://proofwiki.org/wiki/Lower_Sum_of_Refinement | [
"Real Analysis"
] | [
"Definition:Interval/Ordered Set/Closed",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Definition:Lower Darboux Sum"
] | [
"Cardinality of Subset of Finite Set",
"Infimum of Subset"
] |
proofwiki-18959 | Upper Sum of Refinement | Let $\closedint a b$ be a closed interval.
Let $P$ be a finite subdivision of $\closedint a b$.
Let $Q$ be a refinement of $P$.
Then:
:$\map U {f, P} \le \map U {f, Q}$
where $\map U {f, P}$ and $\map U {f, Q}$ denote the upper Darboux sum of $f$ with respect to $P$ and $Q$ respectively. | Write:
:$P = \set {x_0, x_1, \ldots, x_k}$
and:
:$Q = \set {y_0, y_1, \ldots, y_l}$
where:
:$a = x_0 < x_1 < \ldots < x_k = b$
and:
:$a = y_0 < y_1 < \ldots < y_l = b$
Since $P \subseteq Q$, we have $k \le l$ from Cardinality of Subset of Finite Set.
Set:
:$M_i = \sup \set {\map f x : x \in \closedint {x_{i - 1} } {x... | Let $\closedint a b$ be a [[Definition:Closed Interval|closed interval]].
Let $P$ be a [[Definition:Finite Subdivision|finite subdivision]] of $\closedint a b$.
Let $Q$ be a [[Definition:Refinement of Finite Subdivision|refinement]] of $P$.
Then:
:$\map U {f, P} \le \map U {f, Q}$
where $\map U {f, P}$ and $\m... | Write:
:$P = \set {x_0, x_1, \ldots, x_k}$
and:
:$Q = \set {y_0, y_1, \ldots, y_l}$
where:
:$a = x_0 < x_1 < \ldots < x_k = b$
and:
:$a = y_0 < y_1 < \ldots < y_l = b$
Since $P \subseteq Q$, we have $k \le l$ from [[Cardinality of Subset of Finite Set]].
Set:
:$M_i = \sup \set {\map f x : x \in \closedint {... | Upper Sum of Refinement | https://proofwiki.org/wiki/Upper_Sum_of_Refinement | https://proofwiki.org/wiki/Upper_Sum_of_Refinement | [
"Real Analysis"
] | [
"Definition:Interval/Ordered Set/Closed",
"Definition:Subdivision of Interval/Finite",
"Definition:Refinement of Finite Subdivision",
"Definition:Upper Darboux Sum"
] | [
"Cardinality of Subset of Finite Set",
"Supremum of Subset"
] |
proofwiki-18960 | Dirac Delta Distribution is Tempered Distribution | Let $\delta$ be the Dirac delta distribution.
Let $\map {\SS'} \R$ be the tempered distribution space.
Then:
:$\delta \in \map {\SS'} \R$ | Let $\phi \in \map \SS \R$ be a Schwartz test function.
Consider the mapping $\phi \stackrel \delta \longrightarrow \map \phi 0 : \map \SS \R \to \C$.
We have that the Schwartz space is a vector space.
Let $\phi$ be a linear superposition of Schwartz test functions.
Then $\map \phi 0$ is also a linear supperposition ev... | Let $\delta$ be the [[Definition:Dirac Delta Distribution|Dirac delta distribution]].
Let $\map {\SS'} \R$ be the [[Definition:Tempered Distribution Space|tempered distribution space]].
Then:
:$\delta \in \map {\SS'} \R$ | Let $\phi \in \map \SS \R$ be a [[Definition:Schwartz Test Function|Schwartz test function]].
Consider the [[Definition:Mapping|mapping]] $\phi \stackrel \delta \longrightarrow \map \phi 0 : \map \SS \R \to \C$.
We have that the [[Definition:Schwartz Space|Schwartz space]] is a [[Schwartz Space with Pointwise Additio... | Dirac Delta Distribution is Tempered Distribution | https://proofwiki.org/wiki/Dirac_Delta_Distribution_is_Tempered_Distribution | https://proofwiki.org/wiki/Dirac_Delta_Distribution_is_Tempered_Distribution | [
"Tempered Distributions"
] | [
"Definition:Dirac Delta Distribution",
"Definition:Tempered Distribution Space"
] | [
"Definition:Schwartz Test Function",
"Definition:Mapping",
"Definition:Schwartz Space",
"Schwartz Space with Pointwise Addition and Pointwise Scalar Multiplication forms Vector Space",
"Definition:Linear Supperposition",
"Definition:Schwartz Test Function",
"Definition:Linear Supperposition",
"Definit... |
proofwiki-18961 | Condition for Valid Time Indication | Consider an analog clock $C$ with an hour hand $H$ and a minute hand $M$.
Let $\theta \degrees$ be the angle made by the minute hand with respect to twelve o'clock.
Let $\phi \degrees$ be the angle made by the hour hand with respect to twelve o'clock.
Then $C$ displays a '''valid time indication''' {{iff}}:
:$12 \phi \... | Let $T$ be a time of day specified in hours $h$ and minutes $m$, where:
:$1 \le h \le 12$ is an integer
:$0 \le m < 60$ is a real number
whether a.m. or p.m. is immaterial.
From Speed of Minute Hand, $M$ travels $6 \degrees$ per minute.
So at time $m$ minutes after the hour, $\theta = 6 m$.
From Speed of Hour Hand, $H$... | Consider an [[Definition:Analog Clock|analog clock]] $C$ with an [[Definition:Hour Hand|hour hand]] $H$ and a [[Definition:Minute Hand|minute hand]] $M$.
Let $\theta \degrees$ be the [[Definition:Angle|angle]] made by the [[Definition:Minute Hand|minute hand]] with respect to [[Definition:Twelve O'Clock|twelve o'clock... | Let $T$ be a [[Definition:Time of Day|time of day]] specified in [[Definition:Hour|hours]] $h$ and [[Definition:Minute of Time|minutes]] $m$, where:
:$1 \le h \le 12$ is an [[Definition:Integer|integer]]
:$0 \le m < 60$ is a [[Definition:Real Number|real number]]
whether [[Definition:Ante Meridiem|a.m.]] or [[Definitio... | Condition for Valid Time Indication | https://proofwiki.org/wiki/Condition_for_Valid_Time_Indication | https://proofwiki.org/wiki/Condition_for_Valid_Time_Indication | [
"Clocks"
] | [
"Definition:Clock/Analog",
"Definition:Clock/Hour Hand",
"Definition:Clock/Minute Hand",
"Definition:Angle",
"Definition:Clock/Minute Hand",
"Definition:Twelve O'Clock",
"Definition:Angle",
"Definition:Clock/Hour Hand",
"Definition:Twelve O'Clock",
"Definition:Valid Time Indication"
] | [
"Definition:Time of Day",
"Definition:Time/Unit/Hour",
"Definition:Time/Unit/Minute",
"Definition:Integer",
"Definition:Real Number",
"Definition:Ante Meridiem",
"Definition:Post Meridiem",
"Speed of Minute Hand",
"Definition:Time/Unit/Minute",
"Definition:Time/Unit/Minute",
"Definition:Time/Uni... |
proofwiki-18962 | Speed of Minute Hand | Consider an analog clock $C$.
The minute hand of $C$ rotates at $6$ degrees per minute. | It takes one hour, that is $60$ minutes, for the minute hand to go round the dial one time.
That is, in $60$ minutes the minute hand travels $360 \degrees$.
So in $1$ minute, the minute hand travels $\dfrac {360} {60} \degrees$, that is, $6 \degrees$.
{{qed}} | Consider an [[Definition:Analog Clock|analog clock]] $C$.
The [[Definition:Minute Hand|minute hand]] of $C$ rotates at $6$ [[Definition:Degree of Angle|degrees]] per [[Definition:Minute of Time|minute]]. | It takes one [[Definition:Hour|hour]], that is $60$ [[Definition:Minute of Time|minutes]], for the [[Definition:Minute Hand|minute hand]] to go round the [[Definition:Clock Dial|dial]] one time.
That is, in $60$ [[Definition:Minute of Time|minutes]] the [[Definition:Minute Hand|minute hand]] travels $360 \degrees$.
S... | Speed of Minute Hand | https://proofwiki.org/wiki/Speed_of_Minute_Hand | https://proofwiki.org/wiki/Speed_of_Minute_Hand | [
"Clocks"
] | [
"Definition:Clock/Analog",
"Definition:Clock/Minute Hand",
"Definition:Angular Measure/Degree",
"Definition:Time/Unit/Minute"
] | [
"Definition:Time/Unit/Hour",
"Definition:Time/Unit/Minute",
"Definition:Clock/Minute Hand",
"Definition:Clock/Dial",
"Definition:Time/Unit/Minute",
"Definition:Clock/Minute Hand",
"Definition:Time/Unit/Minute",
"Definition:Clock/Minute Hand"
] |
proofwiki-18963 | Speed of Hour Hand | Consider an analog clock $C$.
The hour hand of $C$ rotates at $\dfrac 1 2$ of a degree per minute. | It takes $12$ hours, for the hour hand to go round the dial one time.
That is, in $12$ minutes the hour hand travels $360 \degrees$.
So in $1$ hour, the hour hand travels $\dfrac {360} {12} \degrees$, that is, $30 \degrees$.
So in $1$ minute, the hour hand travels $\dfrac 1 {60} \times 30 \degrees$, that is, $\dfrac 1 ... | Consider an [[Definition:Analog Clock|analog clock]] $C$.
The [[Definition:Hour Hand|hour hand]] of $C$ rotates at $\dfrac 1 2$ of a [[Definition:Degree of Angle|degree]] per [[Definition:Minute of Time|minute]]. | It takes $12$ [[Definition:Hour|hours]], for the [[Definition:Hour Hand|hour hand]] to go round the [[Definition:Clock Dial|dial]] one time.
That is, in $12$ [[Definition:Minute of Time|minutes]] the [[Definition:Hour Hand|hour hand]] travels $360 \degrees$.
So in $1$ [[Definition:Hour|hour]], the [[Definition:Hour H... | Speed of Hour Hand | https://proofwiki.org/wiki/Speed_of_Hour_Hand | https://proofwiki.org/wiki/Speed_of_Hour_Hand | [
"Clocks"
] | [
"Definition:Clock/Analog",
"Definition:Clock/Hour Hand",
"Definition:Angular Measure/Degree",
"Definition:Time/Unit/Minute"
] | [
"Definition:Time/Unit/Hour",
"Definition:Clock/Hour Hand",
"Definition:Clock/Dial",
"Definition:Time/Unit/Minute",
"Definition:Clock/Hour Hand",
"Definition:Time/Unit/Hour",
"Definition:Clock/Hour Hand",
"Definition:Time/Unit/Minute",
"Definition:Clock/Hour Hand"
] |
proofwiki-18964 | Condition for Valid Time Indication/Corollary | Let $\theta \degrees$ be the angle made by the minute hand with respect to twelve o'clock.
Let $\rho \degrees$ be the angle made by the hour hand with respect to the hour just past.
Then $C$ displays a '''valid time indication''' {{iff}}:
:$\rho = \dfrac \theta {12}$ | Follows directly.
{{qed}}
Category:Clocks
3n89lyit4y4z63d6v3s43q859ytvh70 | Let $\theta \degrees$ be the [[Definition:Angle|angle]] made by the [[Definition:Minute Hand|minute hand]] with respect to [[Definition:Twelve O'Clock|twelve o'clock]].
Let $\rho \degrees$ be the [[Definition:Angle|angle]] made by the [[Definition:Hour Hand|hour hand]] with respect to the [[Definition:Hour|hour]] just... | Follows directly.
{{qed}}
[[Category:Clocks]]
3n89lyit4y4z63d6v3s43q859ytvh70 | Condition for Valid Time Indication/Corollary | https://proofwiki.org/wiki/Condition_for_Valid_Time_Indication/Corollary | https://proofwiki.org/wiki/Condition_for_Valid_Time_Indication/Corollary | [
"Clocks"
] | [
"Definition:Angle",
"Definition:Clock/Minute Hand",
"Definition:Twelve O'Clock",
"Definition:Angle",
"Definition:Clock/Hour Hand",
"Definition:Time/Unit/Hour",
"Definition:Valid Time Indication"
] | [
"Category:Clocks"
] |
proofwiki-18965 | Ambiguous Times | Let $T$ be a time of day in $12$-hour clock form.
Then $T$ is an ambiguous time {{iff}}:
:$T = 12:00 + n \times 5 \tfrac 5 {143} \mathrm {min}$
where:
:$n \in \set {1, 2, \ldots, 142}$
:the hour hand and minute hand are pointing in different directions. | Let $T$ be an ambiguous time.
Let $T$ be specified in hours $h$ and minutes $m$, where:
:$1 \le h \le 12$ is an integer
:$0 \le m < 60$ is a real number
whether a.m. or p.m. is immaterial.
At this time $T$:
:let $\theta \degrees$ be the angle made by the minute hand with respect to twelve o'clock
:let $\phi \degrees$ b... | Let $T$ be a [[Definition:Time of Day|time of day]] in [[Definition:Twelve-Hour Clock|$12$-hour clock]] form.
Then $T$ is an [[Definition:Ambiguous Time|ambiguous time]] {{iff}}:
:$T = 12:00 + n \times 5 \tfrac 5 {143} \mathrm {min}$
where:
:$n \in \set {1, 2, \ldots, 142}$
:the [[Definition:Hour Hand|hour hand]] an... | Let $T$ be an [[Definition:Ambiguous Time|ambiguous time]].
Let $T$ be specified in [[Definition:Hour|hours]] $h$ and [[Definition:Minute of Time|minutes]] $m$, where:
:$1 \le h \le 12$ is an [[Definition:Integer|integer]]
:$0 \le m < 60$ is a [[Definition:Real Number|real number]]
whether [[Definition:Ante Meridiem|a... | Ambiguous Times | https://proofwiki.org/wiki/Ambiguous_Times | https://proofwiki.org/wiki/Ambiguous_Times | [
"Ambiguous Times",
"143"
] | [
"Definition:Time of Day",
"Definition:Twelve-Hour Clock",
"Definition:Ambiguous Time",
"Definition:Clock/Hour Hand",
"Definition:Clock/Minute Hand"
] | [
"Definition:Ambiguous Time",
"Definition:Time/Unit/Hour",
"Definition:Time/Unit/Minute",
"Definition:Integer",
"Definition:Real Number",
"Definition:Ante Meridiem",
"Definition:Post Meridiem",
"Definition:Angle",
"Definition:Clock/Minute Hand",
"Definition:Twelve O'Clock",
"Definition:Angle",
... |
proofwiki-18966 | Expectation of Random Variable as Integral with respect to Probability Distribution | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an integrable real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $P_X$ be the probability distribution of $X$.
Then:
:$\ds \expect X = \int_\R x \map {\rd P_X} x$
where $\expect X$ is the expected value of $X$. | From the definition of expectation:
:$\ds \expect X = \int_\Omega X \rd \Pr$
We can write:
:$\ds \int_\Omega X \rd \Pr = \int_\Omega I_\R \circ X \rd \Pr$
where $I_\R$ is the identity map for $\R$.
From the definition of probability distribution, we have:
:$P_X = X_* \Pr$
where $X_* \Pr$ is the pushforward of $\Pr$... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Measure-Integrable Function|integrable]] [[Definition:Real-Valued Random Variable|real-valued random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $P_X$ be the [[Definition:Probability Distrib... | From the definition of [[Definition:Expectation/General Definition|expectation]]:
:$\ds \expect X = \int_\Omega X \rd \Pr$
We can write:
:$\ds \int_\Omega X \rd \Pr = \int_\Omega I_\R \circ X \rd \Pr$
where $I_\R$ is the [[Definition:Identity Mapping|identity map]] for $\R$.
From the definition of [[Definition:... | Expectation of Random Variable as Integral with respect to Probability Distribution | https://proofwiki.org/wiki/Expectation_of_Random_Variable_as_Integral_with_respect_to_Probability_Distribution | https://proofwiki.org/wiki/Expectation_of_Random_Variable_as_Integral_with_respect_to_Probability_Distribution | [
"Expectation"
] | [
"Definition:Probability Space",
"Definition:Integrable Function/Measure Space",
"Definition:Random Variable/Real-Valued",
"Definition:Probability Distribution",
"Definition:Expectation"
] | [
"Definition:Expectation/General Definition",
"Definition:Identity Mapping",
"Definition:Probability Distribution",
"Definition:Pushforward Measure",
"Definition:Borel Sigma-Algebra",
"Integral with respect to Pushforward Measure/Corollary",
"Definition:Integrable Function/Measure Space"
] |
proofwiki-18967 | Expectation of Real-Valued Discrete Random Variable | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a discrete real-valued random variable.
Then $X$ is integrable {{iff}}:
:$\ds \sum_{x \in \Img X} \size x \map \Pr {X = x} < \infty$
in which case:
:$\ds \expect X = \sum_{x \in \Img X} x \map \Pr {X = x}$ | From Characterization of Integrable Functions, we have:
:$X$ is $\Pr$-integrable {{iff}} $\size X$ is $\Pr$-integrable.
That is, $X$ is integrable {{iff}}:
:$\ds \int \size X \rd \Pr < \infty$ | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be a [[Definition:Discrete Random Variable|discrete]] [[Definition:Real-Valued Random Variable|real-valued random variable]].
Then $X$ is [[Definition:Integrable Random Variable|integrable]] {{iff}}:
:$\ds \sum_{x... | From [[Characterization of Integrable Functions]], we have:
:$X$ is [[Definition:Measure-Integrable Function|$\Pr$-integrable]] {{iff}} $\size X$ is [[Definition:Measure-Integrable Function|$\Pr$-integrable]].
That is, $X$ is [[Definition:Integrable Random Variable|integrable]] {{iff}}:
:$\ds \int \size X \rd \Pr <... | Expectation of Real-Valued Discrete Random Variable | https://proofwiki.org/wiki/Expectation_of_Real-Valued_Discrete_Random_Variable | https://proofwiki.org/wiki/Expectation_of_Real-Valued_Discrete_Random_Variable | [
"Discrete Random Variables",
"Expectation",
"Expectation of Discrete Random Variable"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Discrete",
"Definition:Random Variable/Real-Valued",
"Definition:Integrable Random Variable"
] | [
"Characterization of Integrable Functions",
"Definition:Integrable Function/Measure Space",
"Definition:Integrable Function/Measure Space",
"Definition:Integrable Random Variable",
"Definition:Integrable Function/Measure Space"
] |
proofwiki-18968 | Linear Transformation of Real-Valued Random Variable is Real-Valued Random Variable | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.
Let $a$ and $b$ be real numbers.
Then:
:$a X + b$ is a real-valued random variable. | From the definition of a real-valued random variable, we have:
:$X$ is $\Sigma$-measurable.
We want to verify that $a X + b : \Omega \to \R$ is a $\Sigma$-measurable function.
From Pointwise Scalar Multiple of Measurable Function is Measurable, we have:
:$a X$ is $\Sigma$-measurable.
From Constant Function is Measura... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be a [[Definition:Real-Valued Random Variable|real-valued random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $a$ and $b$ be [[Definition:Real Number|real numbers]].
Then:
:$a X + b$ is a [[Definition:Real-... | From the definition of a [[Definition:Real-Valued Random Variable|real-valued random variable]], we have:
:$X$ is [[Definition:Measurable Function|$\Sigma$-measurable]].
We want to verify that $a X + b : \Omega \to \R$ is a [[Definition:Measurable Function|$\Sigma$-measurable function]].
From [[Pointwise Scalar Mul... | Linear Transformation of Real-Valued Random Variable is Real-Valued Random Variable | https://proofwiki.org/wiki/Linear_Transformation_of_Real-Valued_Random_Variable_is_Real-Valued_Random_Variable | https://proofwiki.org/wiki/Linear_Transformation_of_Real-Valued_Random_Variable_is_Real-Valued_Random_Variable | [
"Random Variables"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Real-Valued",
"Definition:Real Number",
"Definition:Random Variable/Real-Valued"
] | [
"Definition:Random Variable/Real-Valued",
"Definition:Measurable Function",
"Definition:Measurable Function",
"Pointwise Scalar Multiple of Measurable Function is Measurable",
"Definition:Measurable Function",
"Constant Function is Measurable",
"Definition:Measurable Function",
"Pointwise Sum of Measu... |
proofwiki-18969 | Singular Random Variable is not Absolutely Continuous | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a singular random variable on $\struct {\Omega, \Sigma, \Pr}$.
Then $X$ is not absolutely continuous. | Let $P_X$ be the probability distribution of $X$.
Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.
Let $\lambda$ be the Lebesgue measure for $\struct {\R, \map \BB \R}$.
From the definition of an absolutely continuous random variable, we have that $X$ is absolutely continuous {{iff}}:
:$P_X$ is absolutely con... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be a [[Definition:Singular Random Variable|singular random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Then $X$ is not [[Definition:Absolutely Continuous Random Variable|absolutely continuous]]. | Let $P_X$ be the [[Definition:Probability Distribution|probability distribution]] of $X$.
Let $\map \BB \R$ be the [[Definition:Borel Sigma-Algebra|Borel $\sigma$-algebra]] on $\R$.
Let $\lambda$ be the [[Definition:Lebesgue Measure|Lebesgue measure]] for $\struct {\R, \map \BB \R}$.
From the definition of an [[De... | Singular Random Variable is not Absolutely Continuous | https://proofwiki.org/wiki/Singular_Random_Variable_is_not_Absolutely_Continuous | https://proofwiki.org/wiki/Singular_Random_Variable_is_not_Absolutely_Continuous | [
"Singular Random Variables"
] | [
"Definition:Probability Space",
"Definition:Singular Random Variable",
"Definition:Absolutely Continuous Random Variable"
] | [
"Definition:Probability Distribution",
"Definition:Borel Sigma-Algebra",
"Definition:Lebesgue Measure",
"Definition:Absolutely Continuous Random Variable",
"Definition:Absolutely Continuous Random Variable",
"Definition:Absolute Continuity/Measure",
"Definition:Borel Sigma-Algebra",
"Definition:Singul... |
proofwiki-18970 | Cumulative Distribution Function is Right-Continuous | :$F_X$ is right-continuous. | Let $x \in \R$.
We show that $F_X$ is right-continuous at $x$.
We use Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals, and will show that:
:for all monotone sequences $\sequence {x_n}_{n \mathop \in \N}$, with $x_n > x$ for each $n$, that converge to $x$ we have:
::$\map {F_X} {x... | :$F_X$ is [[Definition:Right-Continuous Real Function|right-continuous]]. | Let $x \in \R$.
We show that $F_X$ is [[Definition:Right-Continuous Real Function|right-continuous]] at $x$.
We use [[Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals|Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals]], and will show that:
:fo... | Cumulative Distribution Function is Right-Continuous | https://proofwiki.org/wiki/Cumulative_Distribution_Function_is_Right-Continuous | https://proofwiki.org/wiki/Cumulative_Distribution_Function_is_Right-Continuous | [
"Right-Continuous Functions",
"Cumulative Distribution Functions",
"Cumulative Distribution Function is Right-Continuous"
] | [
"Definition:Continuous Real Function/Right-Continuous"
] | [
"Definition:Continuous Real Function/Right-Continuous",
"Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Monotone (Order Theory)/Sequence/Real Sequen... |
proofwiki-18971 | Linear Transformation of Continuous Random Variable is Continuous Random Variable | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $a$ be a non-zero real number.
Let $b$ be a real number.
Let $X$ be a continuous real variable.
Let $F_X$ be the cumulative distribution function of $X$.
Then $a X + b$ is a continuous real variable.
Further, if $a > 0$, the cumulative distribution fun... | From Linear Transformation of Real-Valued Random Variable is Real-Valued Random Variable, $a X + b$ is a real-valued random variable.
Since $X$ is a continuous real variable, we have that:
:$F_X$ is continuous.
We use this fact to show that $F_{a X + b}$ is continuous, showing that $a X + b$ is a continuous real varia... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $a$ be a non-zero [[Definition:Real Number|real number]].
Let $b$ be a [[Definition:Real Number|real number]].
Let $X$ be a [[Definition:Continuous Random Variable|continuous real variable]].
Let $F_X$ be the [[Defin... | From [[Linear Transformation of Real-Valued Random Variable is Real-Valued Random Variable]], $a X + b$ is a [[Definition:Real-Valued Random Variable|real-valued random variable]].
Since $X$ is a [[Definition:Continuous Random Variable|continuous real variable]], we have that:
:$F_X$ is [[Definition:Continuous Real ... | Linear Transformation of Continuous Random Variable is Continuous Random Variable | https://proofwiki.org/wiki/Linear_Transformation_of_Continuous_Random_Variable_is_Continuous_Random_Variable | https://proofwiki.org/wiki/Linear_Transformation_of_Continuous_Random_Variable_is_Continuous_Random_Variable | [
"Continuous Random Variables",
"Cumulative Distribution Functions"
] | [
"Definition:Probability Space",
"Definition:Real Number",
"Definition:Real Number",
"Definition:Random Variable/Continuous",
"Definition:Cumulative Distribution Function",
"Definition:Random Variable/Continuous",
"Definition:Cumulative Distribution Function"
] | [
"Linear Transformation of Real-Valued Random Variable is Real-Valued Random Variable",
"Definition:Random Variable/Real-Valued",
"Definition:Random Variable/Continuous",
"Definition:Continuous Real Function",
"Definition:Continuous Real Function",
"Definition:Random Variable/Continuous",
"Composite of C... |
proofwiki-18972 | Probability of Continuous Random Variable Lying in Singleton Set is Zero | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a continuous real variable on $\struct {\Omega, \Sigma, \Pr}$.
Then, for each $x \in \R$, we have:
:$\map \Pr {X \le x} = \map \Pr {X < x}$
In particular:
:$\map \Pr {X = x} = 0$ | Let $F_X$ be the cumulative distribution function of $X$ so that:
:$\map {F_X} x = \map \Pr {X \le x}$
for each $x \in \R$.
Let $P_X$ be the probability distribution of $X$.
Since $X$ is a continuous real variable, we have:
:$F_X$ is continuous.
From Sequential Continuity is Equivalent to Continuity in the Reals, w... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be a [[Definition:Continuous Random Variable|continuous real variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Then, for each $x \in \R$, we have:
:$\map \Pr {X \le x} = \map \Pr {X < x}$
In particular:
:$\map \Pr... | Let $F_X$ be the [[Definition:Cumulative Distribution Function|cumulative distribution function]] of $X$ so that:
:$\map {F_X} x = \map \Pr {X \le x}$
for each $x \in \R$.
Let $P_X$ be the [[Definition:Probability Distribution|probability distribution]] of $X$.
Since $X$ is a [[Definition:Continuous Random Variab... | Probability of Continuous Random Variable Lying in Singleton Set is Zero | https://proofwiki.org/wiki/Probability_of_Continuous_Random_Variable_Lying_in_Singleton_Set_is_Zero | https://proofwiki.org/wiki/Probability_of_Continuous_Random_Variable_Lying_in_Singleton_Set_is_Zero | [
"Continuous Random Variables",
"Probability of Continuous Random Variable Lying in Singleton Set is Zero"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Continuous"
] | [
"Definition:Cumulative Distribution Function",
"Definition:Probability Distribution",
"Definition:Random Variable/Continuous",
"Definition:Continuous Real Function",
"Sequential Continuity is Equivalent to Continuity in the Reals",
"Definition:Real Number",
"Definition:Sequence",
"Definition:Convergen... |
proofwiki-18973 | Cumulative Distribution Function as Integral of Probability Density Function | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an absolutely continuous random variable.
Let $f_X$ be a probability density function for $X$.
Let $F_X$ be the cumulative distribution function for $X$.
Let $\map \BB \R$ be the Borel $\sigma$-algebra on $\R$.
Let $\lambda$ be the Lebesgue meas... | Let $P_X$ be the probability distribution of $X$.
Since $f_X$ is a probability density function for $X$, $f_X$ is a Radon-Nikodym derivative of $P_X$ with respect to $\lambda$.
Then, we have:
{{begin-eqn}}
{{eqn | l = \map {F_X} x
| r = \map \Pr {X \le x}
| c = {{Defof|Cumulative Distribution Function}}
... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Absolutely Continuous Random Variable|absolutely continuous random variable]].
Let $f_X$ be a [[Definition:Probability Density Function|probability density function]] for $X$.
Let $F_X$ be the [[... | Let $P_X$ be the [[Definition:Probability Distribution|probability distribution]] of $X$.
Since $f_X$ is a [[Definition:Probability Density Function|probability density function]] for $X$, $f_X$ is a [[Definition:Radon-Nikodym Derivative|Radon-Nikodym derivative of $P_X$ with respect to $\lambda$]].
Then, we have: ... | Cumulative Distribution Function as Integral of Probability Density Function | https://proofwiki.org/wiki/Cumulative_Distribution_Function_as_Integral_of_Probability_Density_Function | https://proofwiki.org/wiki/Cumulative_Distribution_Function_as_Integral_of_Probability_Density_Function | [
"Cumulative Distribution Functions",
"Probability Density Functions"
] | [
"Definition:Probability Space",
"Definition:Absolutely Continuous Random Variable",
"Definition:Probability Density Function",
"Definition:Cumulative Distribution Function",
"Definition:Borel Sigma-Algebra",
"Definition:Lebesgue Measure",
"Definition:Lebesgue Integral",
"Definition:Integral of Measure... | [
"Definition:Probability Distribution",
"Definition:Probability Density Function",
"Definition:Radon-Nikodym Derivative"
] |
proofwiki-18974 | Probability of Continuous Random Variable Lying in Singleton Set is Zero/Corollary | Let $C$ be a countable subset of $\R$.
Then:
:$\map \Pr {X \in C} = 0$ | Since $C$ is countable, there exists a sequence $\sequence {x_n}_{n \mathop \in \N}$ of distinct real numbers such that:
:$C = \set {x_n : n \mathop \in \N}$
That is:
:$\ds C = \bigcup_{n \mathop = 1}^\infty \set {x_n}$
where $\set {\set {x_1}, \set {x_2}, \ldots}$ is pairwise disjoint.
We then have:
{{begin-eqn}}
{... | Let $C$ be a [[Definition:Countable Set|countable]] [[Definition:Subset|subset]] of $\R$.
Then:
:$\map \Pr {X \in C} = 0$ | Since $C$ is [[Definition:Countable Set|countable]], there exists a [[Definition:Sequence|sequence]] $\sequence {x_n}_{n \mathop \in \N}$ of distinct [[Definition:Real Number|real numbers]] such that:
:$C = \set {x_n : n \mathop \in \N}$
That is:
:$\ds C = \bigcup_{n \mathop = 1}^\infty \set {x_n}$
where $\set {\... | Probability of Continuous Random Variable Lying in Singleton Set is Zero/Corollary | https://proofwiki.org/wiki/Probability_of_Continuous_Random_Variable_Lying_in_Singleton_Set_is_Zero/Corollary | https://proofwiki.org/wiki/Probability_of_Continuous_Random_Variable_Lying_in_Singleton_Set_is_Zero/Corollary | [
"Probability of Continuous Random Variable Lying in Singleton Set is Zero"
] | [
"Definition:Countable Set",
"Definition:Subset"
] | [
"Definition:Countable Set",
"Definition:Sequence",
"Definition:Real Number",
"Definition:Pairwise Disjoint",
"Definition:Countably Additive Function",
"Probability of Continuous Random Variable Lying in Singleton Set is Zero",
"Category:Probability of Continuous Random Variable Lying in Singleton Set is... |
proofwiki-18975 | Real Sequence with all Subsequences having Convergent Subsequence to Limit Converges to Same Limit | Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence.
Let $x \in \R$.
Suppose that:
:every subsequence $\sequence {x_{n_j} }_{j \in \N}$ of $\sequence {x_n}_{n \in \mathop \N}$ has a subsequence $\sequence {x_{n_{j_k} } }_{k \in \N}$ such that:
::$x_{n_{j_k} } \to x$
Then:
:$x_n \to x$ | {{AimForCont}}, suppose that:
:$x_n$ does not converge to $x$.
Then, there exists some $\epsilon > 0$ such that for every $k \in \N$ there exists $n_k \ge k$ such that:
:$\size {x_{n_k} - x} \ge \epsilon$
Let $\sequence {x_{n_{j_k} } }_{k \in \N}$ be a subsequence of $\sequence {x_{n_j} }_{j \in \N}$.
Then:
:$\size ... | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]].
Let $x \in \R$.
Suppose that:
:every [[Definition:Subsequence|subsequence]] $\sequence {x_{n_j} }_{j \in \N}$ of $\sequence {x_n}_{n \in \mathop \N}$ has a [[Definition:Subsequence|subsequence]] $\sequence {x_{n_{j_k} } }_{k \in \N}$ s... | {{AimForCont}}, suppose that:
:$x_n$ does not [[Definition:Convergent Real Sequence|converge]] to $x$.
Then, there exists some $\epsilon > 0$ such that for every $k \in \N$ there exists $n_k \ge k$ such that:
:$\size {x_{n_k} - x} \ge \epsilon$
Let $\sequence {x_{n_{j_k} } }_{k \in \N}$ be a [[Definition:Subseque... | Real Sequence with all Subsequences having Convergent Subsequence to Limit Converges to Same Limit | https://proofwiki.org/wiki/Real_Sequence_with_all_Subsequences_having_Convergent_Subsequence_to_Limit_Converges_to_Same_Limit | https://proofwiki.org/wiki/Real_Sequence_with_all_Subsequences_having_Convergent_Subsequence_to_Limit_Converges_to_Same_Limit | [
"Limits of Sequences"
] | [
"Definition:Sequence",
"Definition:Subsequence",
"Definition:Subsequence"
] | [
"Definition:Convergent Sequence/Real Numbers",
"Definition:Subsequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Subsequence",
"Definition:Subsequence",
"Definition:Convergent Sequence/Real Numbers",
"Proof by Contradiction",
"Definition:Convergent Sequence/Real Numbers",
"Categor... |
proofwiki-18976 | Sequential Continuity is Equivalent to Continuity in the Reals/Corollary | Let $I$ be a real interval.
Let $x \in I$.
Let $f : I \to \R$ be a real function.
Then $f$ is continuous at $x$ {{iff}}:
:for all monotone sequences $\sequence {x_n}_{n \mathop \in \N}$ converging to $x$ we have:
::$\map f {x_n} \to \map f x$ | === Necessary Condition ===
Suppose $f$ is continuous at $x$, then:
:for all real sequences $\sequence {x_n}_{n \mathop \in \N}$ converging to $x$ we have:
::$\map f {x_n} \to \map f x$
from Sequential Continuity is Equivalent to Continuity in the Reals.
So in particular:
:for all monotone sequences $\sequence {x_n}_... | Let $I$ be a [[Definition:Real Interval|real interval]].
Let $x \in I$.
Let $f : I \to \R$ be a [[Definition:Real Function|real function]].
Then $f$ is [[Definition:Continuous Real Function|continuous]] at $x$ {{iff}}:
:for all [[Definition:Monotone Real Sequence|monotone sequences]] $\sequence {x_n}_{n \mathop ... | === Necessary Condition ===
Suppose $f$ is [[Definition:Continuous Real Function|continuous]] at $x$, then:
:for all [[Definition:Real Sequence|real sequences]] $\sequence {x_n}_{n \mathop \in \N}$ [[Definition:Convergent Real Sequence|converging]] to $x$ we have:
::$\map f {x_n} \to \map f x$
from [[Sequential Co... | Sequential Continuity is Equivalent to Continuity in the Reals/Corollary | https://proofwiki.org/wiki/Sequential_Continuity_is_Equivalent_to_Continuity_in_the_Reals/Corollary | https://proofwiki.org/wiki/Sequential_Continuity_is_Equivalent_to_Continuity_in_the_Reals/Corollary | [
"Sequential Continuity is Equivalent to Continuity in the Reals"
] | [
"Definition:Real Interval",
"Definition:Real Function",
"Definition:Continuous Real Function",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Continuous Real Function",
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Sequential Continuity is Equivalent to Continuity in the Reals",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definition:Mono... |
proofwiki-18977 | Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals | Let $\hointr a b$ be a real interval.
Let $x \in \hointr a b$.
Let $f : \hointr a b \to \R$ be a real function.
Then $f$ is right-continuous at $x$ {{iff}}:
:for all monotone sequences $\sequence {x_n}_{n \mathop \in \N}$, with $x_n > x$ for each $n$, that converge to $x$ we have:
::$\map f {x_n} \to \map f x$ | === Necessary Condition ===
Suppose $f$ is right-continuous at $x$, then:
:for each real sequence $\sequence {x_n}_{n \mathop \in \N}$, with $x_n > x$ for each $n$, converging to $x$ we have:
::$\map f {x_n} \to \map f x$
from Limit of Function by Convergent Sequences: Corollary.
So in particular:
:for all monotone s... | Let $\hointr a b$ be a [[Definition:Real Interval|real interval]].
Let $x \in \hointr a b$.
Let $f : \hointr a b \to \R$ be a [[Definition:Real Function|real function]].
Then $f$ is [[Definition:Right-Continuous Real Function|right-continuous]] at $x$ {{iff}}:
:for all [[Definition:Monotone Real Sequence|monoton... | === Necessary Condition ===
Suppose $f$ is [[Definition:Right-Continuous Real Function|right-continuous]] at $x$, then:
:for each [[Definition:Real Sequence|real sequence]] $\sequence {x_n}_{n \mathop \in \N}$, with $x_n > x$ for each $n$, [[Definition:Convergent Real Sequence|converging]] to $x$ we have:
::$\map f... | Monotonic Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals | https://proofwiki.org/wiki/Monotonic_Sequential_Right-Continuity_is_Equivalent_to_Right-Continuity_in_the_Reals | https://proofwiki.org/wiki/Monotonic_Sequential_Right-Continuity_is_Equivalent_to_Right-Continuity_in_the_Reals | [
"Sequential Right-Continuity is Equivalent to Right-Continuity in the Reals"
] | [
"Definition:Real Interval",
"Definition:Real Function",
"Definition:Continuous Real Function/Right-Continuous",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Definition:Continuous Real Function/Right-Continuous",
"Definition:Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Limit of Function by Convergent Sequences/Corollary",
"Definition:Monotone (Order Theory)/Sequence/Real Sequence",
"Definition:Convergent Sequence/Real Numbers",
"Definitio... |
proofwiki-18978 | Limit of Decreasing Sequence of Unbounded Below Closed Intervals | Let $x \in \R$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a sequence converging to $x$ such that $x_n \ge x$ for each $n \in \N$.
Then:
:$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \hointl {-\infty} x$ | We first show that:
:$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} \subseteq \hointl {-\infty} x$
Let:
:$\ds t \in \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n}$
Then:
:$t \le x_n$
By the Squeeze Theorem, we then have:
:$t \le x$
taking $n \to \infty$.
So $t \in \hointl {-\infty} x$.
{{qed|le... | Let $x \in \R$.
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] [[Definition:Convergent Real Sequence|converging]] to $x$ such that $x_n \ge x$ for each $n \in \N$.
Then:
:$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \hointl {-\infty} x$ | We first show that:
:$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} \subseteq \hointl {-\infty} x$
Let:
:$\ds t \in \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n}$
Then:
:$t \le x_n$
By the [[Squeeze Theorem]], we then have:
:$t \le x$
taking $n \to \infty$.
So $t \in \hointl {-\infty} x$.
... | Limit of Decreasing Sequence of Unbounded Below Closed Intervals | https://proofwiki.org/wiki/Limit_of_Decreasing_Sequence_of_Unbounded_Below_Closed_Intervals | https://proofwiki.org/wiki/Limit_of_Decreasing_Sequence_of_Unbounded_Below_Closed_Intervals | [
"Decreasing Sequences of Sets"
] | [
"Definition:Sequence",
"Definition:Convergent Sequence/Real Numbers"
] | [
"Squeeze Theorem",
"Definition:Set Equality",
"Category:Decreasing Sequences of Sets"
] |
proofwiki-18979 | Sequential Characterization of Limit at Positive Infinity of Real Function | Let $f : \R \to \R$ be a real function.
Let $L$ be a real number.
Then:
:$\ds \lim_{x \to \infty} \map f x = L$
{{iff}}:
:for all real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map f {x_n} \to L$
where:
:$\ds \lim_{x \mathop \to \infty} \map f x$ denotes the limit at $+\infty$ ... | === Necessary Condition ===
Suppose that:
:$\ds \lim_{x \to \infty} \map f x = L$
Let $\sequence {x_n}_{n \mathop \in \N}$ be a real sequence with $x_n \to \infty$.
Let $\epsilon > 0$.
From the definition of limit at infinity, we have:
:there exists $M > 0$ such that for all $x > M$ we have $\size {\map f x - L} < \... | Let $f : \R \to \R$ be a [[Definition:Real Function|real function]].
Let $L$ be a [[Definition:Real Number|real number]].
Then:
:$\ds \lim_{x \to \infty} \map f x = L$
{{iff}}:
:for all [[Definition:Real Sequence|real sequences]] $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map f {x_n... | === Necessary Condition ===
Suppose that:
:$\ds \lim_{x \to \infty} \map f x = L$
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Real Sequence|real sequence]] with $x_n \to \infty$.
Let $\epsilon > 0$.
From the definition of [[Definition:Limit at Infinity|limit at infinity]], we have:
:there exists... | Sequential Characterization of Limit at Positive Infinity of Real Function | https://proofwiki.org/wiki/Sequential_Characterization_of_Limit_at_Positive_Infinity_of_Real_Function | https://proofwiki.org/wiki/Sequential_Characterization_of_Limit_at_Positive_Infinity_of_Real_Function | [
"Limit at Infinity",
"Sequential Characterization of Limit at Positive Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Real Number",
"Definition:Real Sequence",
"Definition:Limit of Real Function/Limit at Infinity/Positive"
] | [
"Definition:Real Sequence",
"Definition:Limit of Real Function/Limit at Infinity/Positive",
"Definition:Unbounded Divergent Sequence/Real Sequence/Positive Infinity",
"Definition:Real Sequence",
"Definition:Real Sequence",
"Definition:Real Sequence",
"Definition:Real Sequence",
"Definition:Real Sequen... |
proofwiki-18980 | Bounds for Cumulative Distribution Function | :$0 \le \map {F_X} x \le 1$ for each $x \in \R$ | From the definition of the cumulative distribution function, we have:
:$\map {F_X} x = \map \Pr {X \le x}$
for each $x \in \R$.
We have:
:$\O \subseteq \set {\omega \in \Omega : \map X \omega \le x} \subseteq \Omega$
So, from Measure is Monotone, we have:
:$\map \Pr \O \le \map \Pr {X \le x} \le \map \Pr \Omega$
Fro... | :$0 \le \map {F_X} x \le 1$ for each $x \in \R$ | From the definition of the [[Definition:Cumulative Distribution Function|cumulative distribution function]], we have:
:$\map {F_X} x = \map \Pr {X \le x}$
for each $x \in \R$.
We have:
:$\O \subseteq \set {\omega \in \Omega : \map X \omega \le x} \subseteq \Omega$
So, from [[Measure is Monotone]], we have:
:$\... | Bounds for Cumulative Distribution Function | https://proofwiki.org/wiki/Bounds_for_Cumulative_Distribution_Function | https://proofwiki.org/wiki/Bounds_for_Cumulative_Distribution_Function | [
"Cumulative Distribution Functions"
] | [] | [
"Definition:Cumulative Distribution Function",
"Measure is Monotone",
"Definition:Probability Measure",
"Category:Cumulative Distribution Functions"
] |
proofwiki-18981 | Cumulative Distribution Function is Increasing | :$F_X$ is an increasing function. | Let $x, y \in \R$ have $x \le y$.
Note that if $\omega \in \Omega$ is such that:
:$\map X \omega \le x$
then:
:$\map X \omega \le y$
so:
:$\set {\omega \in \Omega : \map X \omega \le x} \subseteq \set {\omega \in \Omega : \map X \omega \le y}$
From Measure is Monotone, we then have:
:$\map \Pr {X \le x} \le \map \... | :$F_X$ is an [[Definition:Increasing Real Function|increasing function]]. | Let $x, y \in \R$ have $x \le y$.
Note that if $\omega \in \Omega$ is such that:
:$\map X \omega \le x$
then:
:$\map X \omega \le y$
so:
:$\set {\omega \in \Omega : \map X \omega \le x} \subseteq \set {\omega \in \Omega : \map X \omega \le y}$
From [[Measure is Monotone]], we then have:
:$\map \Pr {X \le x... | Cumulative Distribution Function is Increasing | https://proofwiki.org/wiki/Cumulative_Distribution_Function_is_Increasing | https://proofwiki.org/wiki/Cumulative_Distribution_Function_is_Increasing | [
"Cumulative Distribution Functions"
] | [
"Definition:Increasing/Real Function"
] | [
"Measure is Monotone",
"Definition:Cumulative Distribution Function",
"Definition:Increasing/Real Function"
] |
proofwiki-18982 | Limit of Cumulative Distribution Function at Positive Infinity | :$\ds \lim_{x \mathop \to \infty} \map {F_X} x = 1$
where $\ds \lim_{x \mathop \to \infty} \map {F_X} x$ denotes the limit at $+\infty$ of $F_X$. | From Sequential Characterization of Limit at Positive Infinity of Real Function: Corollary, we aim to show that:
:for all increasing real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to \infty$ we have $\map {F_X} {x_n} \to 1$
at which point we will obtain:
:$\ds \lim_{x \mathop \to \infty} \map {F_X} x ... | :$\ds \lim_{x \mathop \to \infty} \map {F_X} x = 1$
where $\ds \lim_{x \mathop \to \infty} \map {F_X} x$ denotes the [[Definition:Limit at Infinity|limit at $+\infty$]] of $F_X$. | From [[Sequential Characterization of Limit at Positive Infinity of Real Function/Corollary|Sequential Characterization of Limit at Positive Infinity of Real Function: Corollary]], we aim to show that:
:for all [[Definition:Increasing Sequence|increasing]] [[Definition:Real Sequence|real sequences]] $\sequence {x_n}_... | Limit of Cumulative Distribution Function at Positive Infinity | https://proofwiki.org/wiki/Limit_of_Cumulative_Distribution_Function_at_Positive_Infinity | https://proofwiki.org/wiki/Limit_of_Cumulative_Distribution_Function_at_Positive_Infinity | [
"Limit of Cumulative Distribution Function at Positive Infinity",
"Cumulative Distribution Functions",
"Limit at Infinity"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Positive"
] | [
"Sequential Characterization of Limit at Positive Infinity of Real Function/Corollary",
"Definition:Increasing/Sequence",
"Definition:Real Sequence",
"Definition:Increasing/Sequence",
"Definition:Sequence",
"Definition:Increasing Sequence of Sets",
"Definition:Increasing/Sequence",
"Definition:Real Se... |
proofwiki-18983 | Limit of Cumulative Distribution Function at Negative Infinity | :$\ds \lim_{x \mathop \to -\infty} \map {F_X} x = 0$
where $\ds \lim_{x \mathop \to -\infty} \map {F_X} x$ denotes the limit at $-\infty$ of $F_X$. | From Sequential Characterisation of Limit at Minus Infinity of Real Function: Corollary, we aim to show that:
:for all decreasing real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to -\infty$ we have $\map {F_X} {x_n} \to 0$
at which point we will obtain:
:$\ds \lim_{x \mathop \to -\infty} \map {F_X} x =... | :$\ds \lim_{x \mathop \to -\infty} \map {F_X} x = 0$
where $\ds \lim_{x \mathop \to -\infty} \map {F_X} x$ denotes the [[Definition:Limit at Minus Infinity|limit at $-\infty$]] of $F_X$. | From [[Sequential Characterization of Limit at Minus Infinity of Real Function/Corollary|Sequential Characterisation of Limit at Minus Infinity of Real Function: Corollary]], we aim to show that:
:for all [[Definition:Decreasing Sequence|decreasing]] [[Definition:Real Sequence|real sequences]] $\sequence {x_n}_{n \ma... | Limit of Cumulative Distribution Function at Negative Infinity | https://proofwiki.org/wiki/Limit_of_Cumulative_Distribution_Function_at_Negative_Infinity | https://proofwiki.org/wiki/Limit_of_Cumulative_Distribution_Function_at_Negative_Infinity | [
"Limit of Cumulative Distribution Function at Negative Infinity",
"Limit at Minus Infinity",
"Cumulative Distribution Functions"
] | [
"Definition:Limit of Real Function/Limit at Infinity/Negative"
] | [
"Sequential Characterization of Limit at Minus Infinity of Real Function/Corollary",
"Definition:Decreasing/Sequence",
"Definition:Real Sequence",
"Definition:Decreasing/Sequence",
"Definition:Sequence",
"Definition:Decreasing Sequence of Sets",
"Limit of Decreasing Sequence of Unbounded Below Closed In... |
proofwiki-18984 | Absolute Value of Real-Valued Random Variable is Real-Valued Random Variable | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable.
Then $\size X$ is a real-valued random variable. | Since $X$ is a real-valued random variable, $X$ is $\Sigma$-measurable.
From Absolute Value of Measurable Function is Measurable, $\size X$ is $\Sigma$-measurable.
So $\size X$ is a real-valued random variable.
{{qed}}
Category:Random Variables
e13aoey7fx5ems1r5rfwtvrr4sbpnp6 | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be a [[Definition:Real-Valued Random Variable|real-valued random variable]].
Then $\size X$ is a [[Definition:Real-Valued Random Variable|real-valued random variable]]. | Since $X$ is a [[Definition:Real-Valued Random Variable|real-valued random variable]], $X$ is [[Definition:Measurable Function|$\Sigma$-measurable]].
From [[Absolute Value of Measurable Function is Measurable]], $\size X$ is [[Definition:Measurable Function|$\Sigma$-measurable]].
So $\size X$ is a [[Definition:Real-V... | Absolute Value of Real-Valued Random Variable is Real-Valued Random Variable | https://proofwiki.org/wiki/Absolute_Value_of_Real-Valued_Random_Variable_is_Real-Valued_Random_Variable | https://proofwiki.org/wiki/Absolute_Value_of_Real-Valued_Random_Variable_is_Real-Valued_Random_Variable | [
"Random Variables"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Real-Valued",
"Definition:Random Variable/Real-Valued"
] | [
"Definition:Random Variable/Real-Valued",
"Definition:Measurable Function",
"Absolute Value of Measurable Function is Measurable",
"Definition:Measurable Function",
"Definition:Random Variable/Real-Valued",
"Category:Random Variables"
] |
proofwiki-18985 | Positive Part of Real-Valued Random Variable is Real-Valued Random Variable | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable.
Then the positive part $X^+$ of $X$ is a real-valued random variable. | Since $X$ is a real-valued random variable, $X$ is $\Sigma$-measurable.
From Function Measurable iff Positive and Negative Parts Measurable, $X^+$ is $\Sigma$-measurable.
So $X^+$ is a real-valued random variable.
{{qed}}
Category:Random Variables
Category:Positive Parts
pm3g79v0jvdzyrqtauuibf8k4a9yiai | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be a [[Definition:Real-Valued Random Variable|real-valued random variable]].
Then the [[Definition:Positive Part|positive part]] $X^+$ of $X$ is a [[Definition:Real-Valued Random Variable|real-valued random variabl... | Since $X$ is a [[Definition:Real-Valued Random Variable|real-valued random variable]], $X$ is [[Definition:Measurable Function|$\Sigma$-measurable]].
From [[Function Measurable iff Positive and Negative Parts Measurable]], $X^+$ is [[Definition:Measurable Function|$\Sigma$-measurable]].
So $X^+$ is a [[Definition:Rea... | Positive Part of Real-Valued Random Variable is Real-Valued Random Variable | https://proofwiki.org/wiki/Positive_Part_of_Real-Valued_Random_Variable_is_Real-Valued_Random_Variable | https://proofwiki.org/wiki/Positive_Part_of_Real-Valued_Random_Variable_is_Real-Valued_Random_Variable | [
"Positive Parts",
"Random Variables",
"Positive Parts"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Real-Valued",
"Definition:Positive Part",
"Definition:Random Variable/Real-Valued"
] | [
"Definition:Random Variable/Real-Valued",
"Definition:Measurable Function",
"Function Measurable iff Positive and Negative Parts Measurable",
"Definition:Measurable Function",
"Definition:Random Variable/Real-Valued",
"Category:Random Variables",
"Category:Positive Parts"
] |
proofwiki-18986 | Negative Part of Real-Valued Random Variable is Real-Valued Random Variable | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable.
Then the negative part $X^-$ of $X$ is a real-valued random variable. | Since $X$ is a real-valued random variable, $X$ is $\Sigma$-measurable.
From Function Measurable iff Positive and Negative Parts Measurable, $X^-$ is $\Sigma$-measurable.
So $X^-$ is a real-valued random variable.
{{qed}}
Category:Random Variables
Category:Negative Parts
suuv132d4rwhvo0gzxracwbbjw9ppte | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be a [[Definition:Real-Valued Random Variable|real-valued random variable]].
Then the [[Definition:Negative Part|negative part]] $X^-$ of $X$ is a [[Definition:Real-Valued Random Variable|real-valued random variabl... | Since $X$ is a [[Definition:Real-Valued Random Variable|real-valued random variable]], $X$ is [[Definition:Measurable Function|$\Sigma$-measurable]].
From [[Function Measurable iff Positive and Negative Parts Measurable]], $X^-$ is [[Definition:Measurable Function|$\Sigma$-measurable]].
So $X^-$ is a [[Definition:Rea... | Negative Part of Real-Valued Random Variable is Real-Valued Random Variable | https://proofwiki.org/wiki/Negative_Part_of_Real-Valued_Random_Variable_is_Real-Valued_Random_Variable | https://proofwiki.org/wiki/Negative_Part_of_Real-Valued_Random_Variable_is_Real-Valued_Random_Variable | [
"Negative Parts",
"Random Variables",
"Negative Parts"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Real-Valued",
"Definition:Negative Part",
"Definition:Random Variable/Real-Valued"
] | [
"Definition:Random Variable/Real-Valued",
"Definition:Measurable Function",
"Function Measurable iff Positive and Negative Parts Measurable",
"Definition:Measurable Function",
"Definition:Random Variable/Real-Valued",
"Category:Random Variables",
"Category:Negative Parts"
] |
proofwiki-18987 | Probability Distribution is Probability Measure | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\struct {S, \Sigma'}$ be a measurable space.
Let $X$ be a random variable on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.
Let $P_X$ be the probability distribution of $X$.
Then:
:$P_X$ is a probability measure on $\struct {S... | From the definition of probability distribution, we have:
:$P_X = X_* \Pr$
where $X_* \Pr$ is the pushforward $X_* \Pr$ of $\Pr$, under $X$, on $\Sigma'$.
From Pushforward Measure is Measure, we have:
:$P_X$ is a measure.
We then have:
{{begin-eqn}}
{{eqn | l = \map {P_X} S
| r = \map \Pr {X^{-1} \sqbrk S}
| c =... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\struct {S, \Sigma'}$ be a [[Definition:Measurable Space|measurable space]].
Let $X$ be a [[Definition:Random Variable|random variable]] on $\struct {\Omega, \Sigma, \Pr}$ taking values in $\struct {S, \Sigma'}$.
Let ... | From the definition of [[Definition:Probability Distribution|probability distribution]], we have:
:$P_X = X_* \Pr$
where $X_* \Pr$ is the [[Definition:Pushforward Measure|pushforward]] $X_* \Pr$ of $\Pr$, under $X$, on $\Sigma'$.
From [[Pushforward Measure is Measure]], we have:
:$P_X$ is a [[Definition:Measure (... | Probability Distribution is Probability Measure | https://proofwiki.org/wiki/Probability_Distribution_is_Probability_Measure | https://proofwiki.org/wiki/Probability_Distribution_is_Probability_Measure | [
"Probability Distributions"
] | [
"Definition:Probability Space",
"Definition:Measurable Space",
"Definition:Random Variable",
"Definition:Probability Distribution",
"Definition:Probability Measure"
] | [
"Definition:Probability Distribution",
"Definition:Pushforward Measure",
"Pushforward Measure is Measure",
"Definition:Measure (Measure Theory)",
"Definition:Probability Measure",
"Category:Probability Distributions"
] |
proofwiki-18988 | Limit of Decreasing Sequence of Unbounded Below Closed Intervals with Endpoint Tending to Negative Infinity | Let $\sequence {x_n}_{n \mathop \in \N}$ be a decreasing sequence with $x_n \to -\infty$.
Then:
:$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \O$
That is:
:$\hointl {-\infty} {x_n} \downarrow \O$
where $\downarrow$ denotes the limit of decreasing sequence of sets. | {{AimForCont}} suppose that:
:$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} \ne \O$
Let:
:$\ds x \in \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n}$
Then:
:$x \in \hointl {-\infty} {x_n}$ for each $n$.
From the definition of a sequence diverging to $-\infty$:
:there exists $N \in \N$ such that $... | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Decreasing Sequence|decreasing sequence]] with $x_n \to -\infty$.
Then:
:$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} = \O$
That is:
:$\hointl {-\infty} {x_n} \downarrow \O$
where $\downarrow$ denotes the [[Definition:Limit of Decreasing ... | {{AimForCont}} suppose that:
:$\ds \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n} \ne \O$
Let:
:$\ds x \in \bigcap_{n \mathop = 1}^\infty \hointl {-\infty} {x_n}$
Then:
:$x \in \hointl {-\infty} {x_n}$ for each $n$.
From the definition of a [[Definition:Divergent Real Sequence to Negative Infinity|seque... | Limit of Decreasing Sequence of Unbounded Below Closed Intervals with Endpoint Tending to Negative Infinity | https://proofwiki.org/wiki/Limit_of_Decreasing_Sequence_of_Unbounded_Below_Closed_Intervals_with_Endpoint_Tending_to_Negative_Infinity | https://proofwiki.org/wiki/Limit_of_Decreasing_Sequence_of_Unbounded_Below_Closed_Intervals_with_Endpoint_Tending_to_Negative_Infinity | [
"Decreasing Sequences of Sets"
] | [
"Definition:Decreasing/Sequence",
"Definition:Limit of Decreasing Sequence of Sets"
] | [
"Definition:Unbounded Divergent Sequence/Real Sequence/Negative Infinity",
"Proof by Contradiction",
"Category:Decreasing Sequences of Sets"
] |
proofwiki-18989 | Subsequence of Real Sequence Diverging to Positive Infinity Diverges to Positive Infinity | Let $\sequence {x_n}_{n \mathop \in \N}$ be a real sequence with $x_n \to +\infty$.
Let $\sequence {x_{n_j} }_{j \mathop \in \N}$ be a subsequence of $\sequence {x_n}_{n \mathop \in \N}$.
Then:
:$x_{n_j} \to +\infty$ | Let $M > 0$ be a real number.
From the definition of divergence to $+\infty$, there exists $N \in \N$ such that:
:$x_n > M$
for each $n \ge N$.
From Strictly Increasing Sequence of Natural Numbers, we have:
:$n_j \ge j$ for each $j$.
So, we have:
:$x_{n_j} > M$
for each $j \ge N$.
Since $M$ was arbitrary, we have:
... | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Real Sequence|real sequence]] with $x_n \to +\infty$.
Let $\sequence {x_{n_j} }_{j \mathop \in \N}$ be a [[Definition:Subsequence|subsequence]] of $\sequence {x_n}_{n \mathop \in \N}$.
Then:
:$x_{n_j} \to +\infty$ | Let $M > 0$ be a [[Definition:Real Number|real number]].
From the definition of [[Definition:Divergent Real Sequence to Positive Infinity|divergence to $+\infty$]], there exists $N \in \N$ such that:
:$x_n > M$
for each $n \ge N$.
From [[Strictly Increasing Sequence of Natural Numbers]], we have:
:$n_j \ge j$ fo... | Subsequence of Real Sequence Diverging to Positive Infinity Diverges to Positive Infinity | https://proofwiki.org/wiki/Subsequence_of_Real_Sequence_Diverging_to_Positive_Infinity_Diverges_to_Positive_Infinity | https://proofwiki.org/wiki/Subsequence_of_Real_Sequence_Diverging_to_Positive_Infinity_Diverges_to_Positive_Infinity | [
"Divergent Sequences",
"Limits of Sequences"
] | [
"Definition:Real Sequence",
"Definition:Subsequence"
] | [
"Definition:Real Number",
"Definition:Unbounded Divergent Sequence/Real Sequence/Positive Infinity",
"Strictly Increasing Sequence of Natural Numbers",
"Definition:Unbounded Divergent Sequence/Real Sequence/Positive Infinity",
"Category:Divergent Sequences",
"Category:Limits of Sequences"
] |
proofwiki-18990 | Subsequence of Real Sequence Diverging to Negative Infinity Diverges to Negative Infinity | Let $\sequence {x_n}_{n \mathop \in \N}$ be a real sequence with $x_n \to -\infty$.
Let $\sequence {x_{n_j} }_{j \mathop \in \N}$ be a subsequence of $\sequence {x_n}_{n \mathop \in \N}$.
Then:
:$x_{n_j} \to -\infty$ | Let $M > 0$ be a real number.
From the definition of divergence to $-\infty$, there exists $N \in \N$ such that:
:$x_n < -M$
for each $n \ge N$.
From Strictly Increasing Sequence of Natural Numbers, we have:
:$n_j \ge j$ for each $j$.
So, we have:
:$x_{n_j} < -M$
for each $j \ge N$.
Since $M$ was arbitrary, we have:... | Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Real Sequence|real sequence]] with $x_n \to -\infty$.
Let $\sequence {x_{n_j} }_{j \mathop \in \N}$ be a [[Definition:Subsequence|subsequence]] of $\sequence {x_n}_{n \mathop \in \N}$.
Then:
:$x_{n_j} \to -\infty$ | Let $M > 0$ be a [[Definition:Real Number|real number]].
From the definition of [[Definition:Divergent Real Sequence to Negative Infinity|divergence to $-\infty$]], there exists $N \in \N$ such that:
:$x_n < -M$
for each $n \ge N$.
From [[Strictly Increasing Sequence of Natural Numbers]], we have:
:$n_j \ge j$ f... | Subsequence of Real Sequence Diverging to Negative Infinity Diverges to Negative Infinity | https://proofwiki.org/wiki/Subsequence_of_Real_Sequence_Diverging_to_Negative_Infinity_Diverges_to_Negative_Infinity | https://proofwiki.org/wiki/Subsequence_of_Real_Sequence_Diverging_to_Negative_Infinity_Diverges_to_Negative_Infinity | [
"Divergent Sequences",
"Limits of Sequences"
] | [
"Definition:Real Sequence",
"Definition:Subsequence"
] | [
"Definition:Real Number",
"Definition:Unbounded Divergent Sequence/Real Sequence/Negative Infinity",
"Strictly Increasing Sequence of Natural Numbers",
"Definition:Unbounded Divergent Sequence/Real Sequence/Negative Infinity",
"Category:Divergent Sequences",
"Category:Limits of Sequences"
] |
proofwiki-18991 | Area of Equilateral Triangle | Let $T$ be an equilateral triangle.
Let the length of one side of $T$ be $s$.
Let $\AA$ be the area of $T$.
Then:
:$\AA = \dfrac {s^2 \sqrt 3} 4$ | :300px
From Area of Triangle in Terms of Two Sides and Angle:
:$\AA = \dfrac {s^2} 2 \sin 60 \degrees$
From Sine of $60 \degrees$:
:$\sin 60 \degrees = \dfrac {\sqrt 3} 2$
The result follows.
{{qed}}
Category:Areas of Triangles
Category:Equilateral Triangles
gb7x34wc74pnf5o6tyja9n96y41qp6d | Let $T$ be an [[Definition:Equilateral Triangle|equilateral triangle]].
Let the [[Definition:Length (Linear Measure)|length]] of one [[Definition:Side of Polygon|side]] of $T$ be $s$.
Let $\AA$ be the [[Definition:Area|area]] of $T$.
Then:
:$\AA = \dfrac {s^2 \sqrt 3} 4$ | :[[File:Area-of-Equilateral-Triangle.png|300px]]
From [[Area of Triangle in Terms of Two Sides and Angle]]:
:$\AA = \dfrac {s^2} 2 \sin 60 \degrees$
From [[Sine of 60 Degrees|Sine of $60 \degrees$]]:
:$\sin 60 \degrees = \dfrac {\sqrt 3} 2$
The result follows.
{{qed}}
[[Category:Areas of Triangles]]
[[Category:Eq... | Area of Equilateral Triangle | https://proofwiki.org/wiki/Area_of_Equilateral_Triangle | https://proofwiki.org/wiki/Area_of_Equilateral_Triangle | [
"Areas of Triangles",
"Equilateral Triangles"
] | [
"Definition:Triangle (Geometry)/Equilateral",
"Definition:Linear Measure/Length",
"Definition:Polygon/Side",
"Definition:Area"
] | [
"File:Area-of-Equilateral-Triangle.png",
"Area of Triangle in Terms of Two Sides and Angle",
"Sine of 60 Degrees",
"Category:Areas of Triangles",
"Category:Equilateral Triangles"
] |
proofwiki-18992 | Sequential Characterization of Limit at Minus Infinity of Real Function | Let $f : \R \to \R$ be a real function.
Let $L$ be a real number.
Then:
:$\ds \lim_{x \to -\infty} \map f x = L$
{{iff}}:
:for all real sequences $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to -\infty$ we have $\map f {x_n} \to L$
where:
:$\ds \lim_{x \mathop \to -\infty} \map f x$ denotes the limit at $-\inft... | === Necessary Condition ===
Suppose that:
:$\ds \lim_{x \to -\infty} \map f x = L$
Let $\sequence {x_n}_{n \mathop \in \N}$ be a real sequence with $x_n \to -\infty$.
Let $\epsilon > 0$.
From the definition of limit at $-\infty$, we have:
:there exists $M > 0$ such that for all $x < -M$ we have $\size {\map f x - L}... | Let $f : \R \to \R$ be a [[Definition:Real Function|real function]].
Let $L$ be a [[Definition:Real Number|real number]].
Then:
:$\ds \lim_{x \to -\infty} \map f x = L$
{{iff}}:
:for all [[Definition:Real Sequence|real sequences]] $\sequence {x_n}_{n \mathop \in \N}$ with $x_n \to -\infty$ we have $\map f {x... | === Necessary Condition ===
Suppose that:
:$\ds \lim_{x \to -\infty} \map f x = L$
Let $\sequence {x_n}_{n \mathop \in \N}$ be a [[Definition:Real Sequence|real sequence]] with $x_n \to -\infty$.
Let $\epsilon > 0$.
From the definition of [[Definition:Limit at Minus Infinity|limit at $-\infty$]], we have:
:the... | Sequential Characterization of Limit at Minus Infinity of Real Function | https://proofwiki.org/wiki/Sequential_Characterization_of_Limit_at_Minus_Infinity_of_Real_Function | https://proofwiki.org/wiki/Sequential_Characterization_of_Limit_at_Minus_Infinity_of_Real_Function | [
"Limit at Minus Infinity",
"Sequential Characterization of Limit at Minus Infinity of Real Function"
] | [
"Definition:Real Function",
"Definition:Real Number",
"Definition:Real Sequence",
"Definition:Limit of Real Function/Limit at Infinity/Negative"
] | [
"Definition:Real Sequence",
"Definition:Limit of Real Function/Limit at Infinity/Negative",
"Definition:Unbounded Divergent Sequence/Real Sequence/Negative Infinity",
"Definition:Real Sequence",
"Definition:Real Sequence",
"Definition:Real Sequence",
"Definition:Real Sequence",
"Definition:Real Sequen... |
proofwiki-18993 | Lebesgue Infinity-Space is Subset of Tempered Distribution Space | Let $\map {L^\infty} \R$ be the Lebesgue infinity-space.
Let $\map {\SS'} \R$ be the tempered distribution space.
Then in the distributional sense it holds that:
:$\map {L^\infty} \R \subseteq \map {\SS'} \R$
That is, every Schwartz distribution defined by an element of $\map {L^\infty} \R$ is a tempered distribution. | Let $f \in \map {L^\infty} \R$.
Let $\phi \in \map \SS \R$ be a Schwartz test function.
Let $T_f : \map \SS \R \to \R$ be a mapping such that:
:$\ds \map {T_f} \phi = \int_\R \map f x \map \phi x \rd x$
Then:
{{begin-eqn}}
{{eqn | l = \size {\map {T_f} \phi}
| r = \size {\int_\R \map f x \map \phi x \rd x}
}}
{{e... | Let $\map {L^\infty} \R$ be the [[Definition:Lebesgue Infinity-Space|Lebesgue infinity-space]].
Let $\map {\SS'} \R$ be the [[Definition:Tempered Distribution Space|tempered distribution space]].
Then in the [[Definition:Tempered Distribution|distributional]] sense it holds that:
:$\map {L^\infty} \R \subseteq \map... | Let $f \in \map {L^\infty} \R$.
Let $\phi \in \map \SS \R$ be a [[Definition:Schwartz Test Function|Schwartz test function]].
Let $T_f : \map \SS \R \to \R$ be a [[Definition:Mapping|mapping]] such that:
:$\ds \map {T_f} \phi = \int_\R \map f x \map \phi x \rd x$
Then:
{{begin-eqn}}
{{eqn | l = \size {\map {T_f} \... | Lebesgue Infinity-Space is Subset of Tempered Distribution Space | https://proofwiki.org/wiki/Lebesgue_Infinity-Space_is_Subset_of_Tempered_Distribution_Space | https://proofwiki.org/wiki/Lebesgue_Infinity-Space_is_Subset_of_Tempered_Distribution_Space | [
"Tempered Distributions"
] | [
"Definition:Lebesgue Space/L-Infinity",
"Definition:Tempered Distribution Space",
"Definition:Tempered Distribution",
"Definition:Schwartz Distribution",
"Definition:Element",
"Definition:Tempered Distribution"
] | [
"Definition:Schwartz Test Function",
"Definition:Mapping",
"Definition:Sequence",
"Definition:Zero-Limit Sequence in Schwartz Space",
"Definition:Zero Mapping",
"Definition:Schwartz Test Function"
] |
proofwiki-18994 | Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ and $Y$ be real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$.
Let $\alpha$ and $\beta$ be real numbers.
Then:
:$\alpha X + \beta Y$ is a real-valued random variable. | Since $X$ and $Y$ are real-valued random variables, we have:
:$X$ and $Y$ are $\Sigma$-measurable functions.
From Pointwise Scalar Multiple of Measurable Function is Measurable, we have:
:$\alpha X$ and $\beta Y$ are $\Sigma$-measurable.
From Pointwise Sum of Measurable Functions is Measurable, we have:
:$\alpha X +... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ and $Y$ be [[Definition:Real-Valued Random Variable|real-valued random variables]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $\alpha$ and $\beta$ be [[Definition:Real Number|real numbers]].
Then:
:$\alpha X + \beta ... | Since $X$ and $Y$ are [[Definition:Real-Valued Random Variable|real-valued random variables]], we have:
:$X$ and $Y$ are [[Definition:Measurable Real-Valued Function|$\Sigma$-measurable functions]].
From [[Pointwise Scalar Multiple of Measurable Function is Measurable]], we have:
:$\alpha X$ and $\beta Y$ are [[De... | Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable | https://proofwiki.org/wiki/Linear_Combination_of_Real-Valued_Random_Variables_is_Real-Valued_Random_Variable | https://proofwiki.org/wiki/Linear_Combination_of_Real-Valued_Random_Variables_is_Real-Valued_Random_Variable | [
"Random Variables",
"Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable"
] | [
"Definition:Probability Space",
"Definition:Random Variable/Real-Valued",
"Definition:Real Number",
"Definition:Random Variable/Real-Valued"
] | [
"Definition:Random Variable/Real-Valued",
"Definition:Measurable Function/Real-Valued Function",
"Pointwise Scalar Multiple of Measurable Function is Measurable",
"Definition:Measurable Function/Real-Valued Function",
"Pointwise Sum of Measurable Functions is Measurable",
"Definition:Measurable Function/R... |
proofwiki-18995 | Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable/General Result | Let $n \in \N$.
Let $\sequence {X_i}_{i \mathop \in \N}$ be a sequence of real-valued random variables on $\struct {\Omega, \Sigma, \Pr}$.
Let $\sequence {\alpha_i}_{i \mathop \in \N}$ be a sequence of real numbers.
Then:
:$\ds \sum_{i \mathop = 1}^n \alpha_i X_i$ is a real-valued random variable. | We proceed by induction.
For all $n \in \N$ let $\map P n$ be the proposition:
:$\ds \sum_{i \mathop = 1}^n \alpha_i X_i$ is $\Sigma$-measurable. | Let $n \in \N$.
Let $\sequence {X_i}_{i \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real-Valued Random Variable|real-valued random variables]] on $\struct {\Omega, \Sigma, \Pr}$.
Let $\sequence {\alpha_i}_{i \mathop \in \N}$ be a [[Definition:Sequence|sequence]] of [[Definition:Real Number... | We proceed by [[Principle of Mathematical Induction|induction]].
For all $n \in \N$ let $\map P n$ be the [[Definition:Proposition|proposition]]:
:$\ds \sum_{i \mathop = 1}^n \alpha_i X_i$ is [[Definition:Measurable Function|$\Sigma$-measurable]]. | Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable/General Result | https://proofwiki.org/wiki/Linear_Combination_of_Real-Valued_Random_Variables_is_Real-Valued_Random_Variable/General_Result | https://proofwiki.org/wiki/Linear_Combination_of_Real-Valued_Random_Variables_is_Real-Valued_Random_Variable/General_Result | [
"Linear Combination of Real-Valued Random Variables is Real-Valued Random Variable"
] | [
"Definition:Sequence",
"Definition:Random Variable/Real-Valued",
"Definition:Sequence",
"Definition:Real Number",
"Definition:Random Variable/Real-Valued"
] | [
"Principle of Mathematical Induction",
"Definition:Proposition",
"Definition:Measurable Function"
] |
proofwiki-18996 | Absolutely Continuous Random Variable is Continuous | Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be an absolutely continuous random variable on $\struct {\Omega, \Sigma, \Pr}$.
Then $X$ is a continuous random variable. | Let $F_X$ be the cumulative distribution function for $X$.
From the definition of an absolutely continuous random variable, we have:
:$F_X$ is absolutely continuous.
From Absolutely Continuous Real Function is Continuous, we then have:
:$F_X$ is continuous.
Then, from the definition of a continuous random variable, ... | Let $\struct {\Omega, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $X$ be an [[Definition:Absolutely Continuous Random Variable|absolutely continuous random variable]] on $\struct {\Omega, \Sigma, \Pr}$.
Then $X$ is a [[Definition:Continuous Random Variable|continuous random variable]]... | Let $F_X$ be the [[Definition:Cumulative Distribution Function|cumulative distribution function]] for $X$.
From the definition of an [[Definition:Absolutely Continuous Random Variable|absolutely continuous random variable]], we have:
:$F_X$ is [[Definition:Absolutely Continuous Real Function|absolutely continuous]]... | Absolutely Continuous Random Variable is Continuous | https://proofwiki.org/wiki/Absolutely_Continuous_Random_Variable_is_Continuous | https://proofwiki.org/wiki/Absolutely_Continuous_Random_Variable_is_Continuous | [
"Absolutely Continuous Random Variables",
"Continuous Random Variables"
] | [
"Definition:Probability Space",
"Definition:Absolutely Continuous Random Variable",
"Definition:Random Variable/Continuous"
] | [
"Definition:Cumulative Distribution Function",
"Definition:Absolutely Continuous Random Variable",
"Definition:Absolute Continuity/Real Function",
"Absolutely Continuous Real Function is Continuous",
"Definition:Continuous Real Function",
"Definition:Random Variable/Continuous",
"Definition:Random Varia... |
proofwiki-18997 | Fourier Transform of Tempered Distribution on 1-Lebesgue Space equals Tempered Distribution of Fourier Transform of defining Function | Let $\map {L^1} \R$ be the Lebesgue $1$-space.
Let $f \in \map {L^1} \R$.
Let $T_f \in \map {\SS'} \R$ be a tempered distribution associated with $f$.
Then:
:$\hat T_f = T_{\hat f}$
where:
:$\hat T_f$ denotes the Fourier transform of the tempered distribution $T_f$
:$\hat f$ denotes the Fourier transform of the real f... | Let $\phi \in \map \SS \R$ be a Schwartz test function.
{{begin-eqn}}
{{eqn | l = \map { {\hat T}_f} \phi
| r = \map {T_f} {\hat \phi}
| c = {{Defof|Fourier Transform of Tempered Distribution}}
}}
{{eqn | r = \int_\R \map f x \map {\hat \phi} x \rd x
| c = {{Defof|Tempered Distribution}}
}}
{{eqn | r ... | Let $\map {L^1} \R$ be the [[Definition:Lebesgue Space|Lebesgue $1$-space]].
Let $f \in \map {L^1} \R$.
Let $T_f \in \map {\SS'} \R$ be a [[Definition:Tempered Distribution|tempered distribution]] [[Lebesgue 1-Space is Subset of Tempered Distribution Space|associated]] with $f$.
Then:
:$\hat T_f = T_{\hat f}$
wher... | Let $\phi \in \map \SS \R$ be a [[Definition:Schwartz Test Function|Schwartz test function]].
{{begin-eqn}}
{{eqn | l = \map { {\hat T}_f} \phi
| r = \map {T_f} {\hat \phi}
| c = {{Defof|Fourier Transform of Tempered Distribution}}
}}
{{eqn | r = \int_\R \map f x \map {\hat \phi} x \rd x
| c = {{Defo... | Fourier Transform of Tempered Distribution on 1-Lebesgue Space equals Tempered Distribution of Fourier Transform of defining Function | https://proofwiki.org/wiki/Fourier_Transform_of_Tempered_Distribution_on_1-Lebesgue_Space_equals_Tempered_Distribution_of_Fourier_Transform_of_defining_Function | https://proofwiki.org/wiki/Fourier_Transform_of_Tempered_Distribution_on_1-Lebesgue_Space_equals_Tempered_Distribution_of_Fourier_Transform_of_defining_Function | [
"Tempered Distributions",
"Fourier Transforms"
] | [
"Definition:Lebesgue Space",
"Definition:Tempered Distribution",
"Lebesgue 1-Space is Subset of Tempered Distribution Space",
"Definition:Fourier Transform of Tempered Distribution",
"Definition:Tempered Distribution",
"Definition:Fourier Transform/Real Function",
"Definition:Real Function"
] | [
"Definition:Schwartz Test Function",
"Fubini's Theorem",
"Fourier Transform of 1-Lebesgue Space Function is Bounded",
"Lebesgue Infinity-Space is Subset of Tempered Distribution Space",
"Definition:Tempered Distribution",
"Definition:Tempered Distribution"
] |
proofwiki-18998 | Random Vector is Random Variable | Let $n \in \N$.
Let $\struct {X, \Sigma, \Pr}$ be a probability space.
Let $\struct {S_1, \Sigma_1}, \struct {S_2, \Sigma_2}, \ldots, \struct {S_n, \Sigma_n}$ be measurable spaces.
Let:
:$\ds S = \prod_{i \mathop = 1}^n S_i$
Let:
:$\ds \Sigma' = \bigotimes_{i \mathop = 1}^n \Sigma_i$
where $\ds \bigotimes_{i \matho... | Note that from the definition of product $\sigma$-algebra: finite case, we have:
:$\ds \Sigma' = \map \sigma {\set {\prod_{i \mathop = 1}^n S_i : S_i \in \Sigma_i \text { for each } i \in \set {1, 2, \ldots, n} } }$
where $\sigma$ denotes the $\sigma$-algebra generated by a collection of subsets.
We aim to apply Mappi... | Let $n \in \N$.
Let $\struct {X, \Sigma, \Pr}$ be a [[Definition:Probability Space|probability space]].
Let $\struct {S_1, \Sigma_1}, \struct {S_2, \Sigma_2}, \ldots, \struct {S_n, \Sigma_n}$ be [[Definition:Measurable Space|measurable spaces]].
Let:
:$\ds S = \prod_{i \mathop = 1}^n S_i$
Let:
:$\ds \Sigma' = ... | Note that from the definition of [[Definition:Product Sigma-Algebra/Finite Case|product $\sigma$-algebra: finite case]], we have:
:$\ds \Sigma' = \map \sigma {\set {\prod_{i \mathop = 1}^n S_i : S_i \in \Sigma_i \text { for each } i \in \set {1, 2, \ldots, n} } }$
where $\sigma$ denotes the [[Definition:Sigma-Algebra... | Random Vector is Random Variable | https://proofwiki.org/wiki/Random_Vector_is_Random_Variable | https://proofwiki.org/wiki/Random_Vector_is_Random_Variable | [
"Random Vectors",
"Random Variables"
] | [
"Definition:Probability Space",
"Definition:Measurable Space",
"Definition:Product Sigma-Algebra/Finite Case",
"Definition:Random Variable",
"Definition:Function",
"Definition:Random Variable"
] | [
"Definition:Product Sigma-Algebra/Finite Case",
"Definition:Sigma-Algebra Generated by Collection of Subsets",
"Mapping Measurable iff Measurable on Generator",
"Definition:Measurable Mapping",
"Sigma-Algebra Closed under Finite Intersection",
"Mapping Measurable iff Measurable on Generator",
"Definitio... |
proofwiki-18999 | Fourier Transform of Dirac Delta Distribution | Let $\delta \in \map {\SS'} \R$ be the Dirac delta distribution.
Let $\mathbf 1 : \map \SS \R \to \R$ be the constant tempered distribution such that for all $\phi \in \map \SS \R$ we have:
:$\ds \map {\mathbf 1} \phi = \int_{-\infty}^\infty 1 \cdot \map \phi x \rd x$
Then in the distributional sense it holds that:
:$\... | Let $\phi \in \map \SS \R$ be a Schwartz test function.
Then:
{{begin-eqn}}
{{eqn | l = \map {\hat \delta} \phi
| r = \map \delta {\hat \phi}
| c = {{Defof|Fourier Transform of Tempered Distribution}}
}}
{{eqn | r = \map {\hat \phi} 0
| c = {{Defof|Tempered Dirac Delta Distribution}}
}}
{{eqn | r = \i... | Let $\delta \in \map {\SS'} \R$ be the [[Definition:Tempered Dirac Delta Distribution|Dirac delta distribution]].
Let $\mathbf 1 : \map \SS \R \to \R$ be the [[Definition:Constant Tempered Distribution|constant tempered distribution]] such that for all $\phi \in \map \SS \R$ we have:
:$\ds \map {\mathbf 1} \phi = \in... | Let $\phi \in \map \SS \R$ be a [[Definition:Schwartz Test Function|Schwartz test function]].
Then:
{{begin-eqn}}
{{eqn | l = \map {\hat \delta} \phi
| r = \map \delta {\hat \phi}
| c = {{Defof|Fourier Transform of Tempered Distribution}}
}}
{{eqn | r = \map {\hat \phi} 0
| c = {{Defof|Tempered Dira... | Fourier Transform of Dirac Delta Distribution | https://proofwiki.org/wiki/Fourier_Transform_of_Dirac_Delta_Distribution | https://proofwiki.org/wiki/Fourier_Transform_of_Dirac_Delta_Distribution | [
"Tempered Distributions",
"Fourier Transforms"
] | [
"Definition:Tempered Dirac Delta Distribution",
"Definition:Constant Tempered Distribution",
"Definition:Tempered Distribution",
"Definition:Fourier Transform of Tempered Distribution"
] | [
"Definition:Schwartz Test Function"
] |
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